Spain

PHYSICAL REVIEW A, VOLUME 64, 013201

Electronic excitation energies of ZniSi clusters

Jon M. Matxain,1 Arantxa Irigoras,2 Joseph E. Fowler,1 and Jesus M. Ugalde1

1Kimika Fakultatea, Euskal Herriko Unibertsitatea, P.K. 1072, 20018 Donostia, Euskadi, Spain2Humanitateak eta Hezkuntza Zientzien Fakultatea, Mondragon Unibersitatea, Dorleta auzoa, z/g, 20540, Eskoriatza, Euskadi,

~Received 20 October 2000; revised manuscript received 13 February 2001; published 13 June 2001!

Time-dependent density-functional theory~TDDFT! excitation energies are calculated for ZniSi globalminima clusters,i 51 –9. The geometry of the global minima is ringlike fori 51 –5 and three-dimensional~3D! spheroidlike fori 56 –9. In general, the calculated excitations happen from nonbondingp orbitals ofsulfur. These orbitals are perpendicular to the molecular plane in the case of the rings, and normal to thespheroid surface for 3D clusters. The calculated excitation energies are larger for ringlike clusters as comparedto 3D ones, with the excitation energies of the latter structures lying close to the visible spectrum. Thedifference between Kohn-Sham eigenvalues of the orbitals involved in the electronic excitations studied havealso been compared with the TDDFT results of the corresponding excitations for two approximate densityfunctionals, i.e., MPW1PW91 and B3LYP, the latter being more accurate. The B3LYP excitation energiescalculated as the difference between Kohn-Sham eigenvalues of the orbitals involved in the excitation havebeen found to be only 0.30–0.40 eV too high for the smaller 3D-like clusters. Moreover, they approach theTDDFT value as the cluster size increases. Therefore, this might be a practical approach to estimate excitationenergies of large ZniSi clusters.

DOI: 10.1103/PhysRevA.64.013201 PACS number~s!: 36.40.2c

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I. INTRODUCTION

The end of the 20th century has seen a dramatic increin interest in II-VI compound alloys. Due to their specisemiconductor properties, these compounds are suitableapplications such as photovoltaic solar cells@1–10#, opticalsensitizers@11#, photocatalysts@12,13#, and quantum device@14#. As can be appreciated, such important applicabilityled to extensive investigation, both experimental and theoical @15–24#.

Semiconduction arises from the existence of no~nule!band gaps that may be gained by taking from the envirment. One manner of achieving this is absorbing the eneof the photons that come from the sun. This is basicallyphysical phenomenon occurring in solar cells. A good marial for photovoltaic solar cells must, therefore, have a bagap within the range of photon energies emitted fromsun. Most of the energy coming from the sun lies within tvisible spectrum. Even the remaining radiation also lnearby, both in the infrared and the ultraviolet. Additionalsolar cells consist of ann-type semiconductor and ap-typesemiconductor, in order to obtain two surfaces with a diffent electric potential to generate electricity. The II-VI compound semiconductors fulfil these requirements. Their bgaps range from 1.45 eV for CdTe to 3.66 eV for ZnS, aall these compounds present eitherp- or n-type semiconduc-tion.

On the other side, since the recent development of natechnology, clusters and nanostructures have become iesting in themselves. They might be experimentally charterized, hence they are no longer only a theoretical toostudy infinite systems. Nowadays, clusters have many apcations and consequently the literature in the field is growrapidly. Many experimental@25–30# and theoretical@31–36#studies have been reported concerning clusters of varcompositions, which have important and interesting appli

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tions. A nice example of this are the fullerenes, carbon sproids discovered by Curl, Smalley, and Kroto, which alhave photovoltaic applications@37,38#.

The study of small II-VI clusters appears promisintherefore. Global and local minima of ZniSi , i 51 –9 @39#,and ZniOi , i 51 –9, clusters@40# have been characterizeearlier at the Becke 3 Lee-Yang-Perdew combined withrelativistic pseudopotentials of Stevens, Krauss, Basch,Jasien@B3LYP-SKBJ~d!# level of theory. Concomitantly, ina previous study@41#, both configuration-interation single~CIS! and time-dependent density-functional theory~TD-DFT! methods were tested for ZniSi , i 51 –3. In both casesthe SKBJ relativistic pseudopotentials with shared exponewere used and the basis set was systematically augmentfind the most proper one. Multireference configuratiointeraction~MR-CI! calculations were performed to servea reference due to the lack of experimental data. It was ccluded that the TDDFT-SKBJ(1sp2d2 f ) level of theorywas the most appropriate.

In this paper, a full investigation of the electronic excittion energies of ZniSi , i 51 –9, global minima clusters acharacterized in Ref.@39#, is presented. The motivation fothis study was to estimate the absorption properties of thsmall clusters, in order to assess whether they are suitablconstructing highly efficient solar cells.

II. METHOD

The optimized structures of the ZniSi clusters used in thiswork were optimized earlier@39# at the B3LYP/SKBJ(d)level of theory.

The electronic excitation energies have been calculawith the time-dependent density-functional theory~TDDFT!@42–44# as recommended in our previous analysis@41# of theexcitation energies of small clusters. For the TDDFT calc

©2001 The American Physical Society01-1

MATXAIN, IRIGORAS, FOWLER, AND UGALDE PHYSICAL REVIEW A 64 013201

TABLE I. SKBJ(d) and SKBJ(1sp2d2 f ) excitation energies~eV! and oscillator strengths of small ZniSi , i 51 –5. D(DE) is thedifference between SKBJ(1sp2d2 f ) DE and SKBJ(d) DE. Similarly for D f .

Transition SKBJ(1sp2d2 f ) SKBJ(1d)Val. Conf Orbitals States DE f DE f D(DE) D f

Zn1S1 2d2 4s2 5s2 5p4 5p→6s 1S→1P 0.81 0.002 0.76 0.001 0.05 0.001LUMO:6s 5s→6s 1S→1S 3.96 0.062 3.99 0.060 0.03 0.002

Zn2S2 5b2u2 3b3g

2 3b3u2 4b1u

2 2b2g2 2b2g→7ag

1Ag→1B2g 2.13 forbidden 2.06 forbidden 0.07LUMO:7ag 4b1u→7ag

1Ag→1B1u 2.90 0.015 2.87 0.014 0.03 0.0013b3u→7ag

1Ag→1B3u 2.97 0.020 2.93 0.015 0.04 0.005

Zn3S3 14a12 12b2

2 6b12 5a2

2 7b12 7b1→15a1

1A1→1B1 3.60 forbidden 3.61 forbidden 0.01LUMO: 15a1 5a2→15a1

1A1→1A2 3.60 forbidden 3.61 forbidden 0.016b1→15a1

1A1→1B1 4.32 0.038 4.33 0.042 0.01 0.004

Zn4S4 4b2g2 3a2u

2 7eg2 8eg

2 2b1u2 2b1u→7a1g

1A1g→1B1u 3.74 forbidden 3.72 forbidden 0.02LUMO:7a1g8a1g 3a2u→7a1g

1A1g→1A2u 4.66 0.005 4.65 0.011 0.01 0.0063a2u→8a1g

1A1g→1A2u 4.92 0.088 4.95 0.096 0.03 0.008

Zn5S5 33a82 29a92 34a82 35a82 30a92 30a9→36a8 1A8→1A9 4.01 0.001 3.97 0.001 0.04LUMO:36a8 35a8→36a8 1A8→1A8 4.21 0.001 4.18 0.001 0.03

30a9→37a8 1A8→1A9 4.39 0.001 4.39 0.007 0 0.006

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lations, the hybrid Becke-style one-parameter functionaling modified Perdew-Wang exchange and Perdew-Wangcorrelation~MPW1PW91! @45# was used.

The relativistic compact effective core potentials withshared-exponent basis set@46# of Stevens, Krauss, Baschand Jasien~SKBJ! were used as the starting basis set. Wdemonstrated previously@41# that polarization and diffusionfunctions were needed in order to yield accurate results,we showed that the SKBJ(1sp2d2 f ) basis, where there weradded ones andp, two d, and twof functions, was the ideaone for an accurate though affordable calculation of the etronic excitation energies of small ZniSi clusters. Neverthe-less, the size of the basis set makes the calculation ofexcitation energies of large clusters prohibitive. Hencefothe smaller SKBJ(d) basis set was chosen for these largclusters. All the exponents of the expanded SKBJ basisthe same as used in@41#.

All the calculations were carried out with theGAUSSIAN98

@47# package.

III. RESULTS AND DISCUSSION

In Ref. @39#, it was concluded that the global minima osmall ZniSi , i 51 –5, clusters were ring structures while thglobal minima for larger clusters,i 56 –9, were three-dimensional spheroids. Ringlike structures will constitute ogroup 1 and the three-dimensional structures will be in gro2, in an attempt to present the results in a more understable way.

For clusters of group 1, the previously mentionSKBJ(1sp2d2 f ) basis was used. However, as indicatedthe preceding section, for clusters of group 2 this basis is

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large. Henceforth, the smaller SKBJ(d) basis set was chosefor these larger clusters. However, for the clusters of gro1, we have obtained the electronic excitation energies wboth SKBJ(d) and SKBJ(1sp2d2 f ) basis sets. This allowsus to assess the performance of the smaller basis setconsequently to highlight the limitations of the SKBJ(d) ba-sis set with respect to SKBJ(1sp2d2 f ).

A. TDDFT excitation energies of ZniSi , iÄ1–9

1. Group 1

Table I collects the SKBJ(d) and SKBJ(1sp2d2 f ) exci-tation energies and oscillator strengths for the clustersgroup 1, namely ZniSi , i 51 –5, which are depicted in Fig1.

The lowest excitation of Zn1S1 occurs when an electron iexcited from the highest occupied molecular orbi~HOMO! to the lowest unoccupied molecular orbit~LUMO!, yielding a 1P excited state. This transition need0.81 eV to happen, and has a small oscillator strengthf50.002. The other calculated transition involves the1Sground state and an excited state of the same symmetry.the electron is excited from the inner 5s orbital to theLUMO. The excitation energy is 3.96 eV, 3.15 eV largthan the1S→1P excitation, but the oscillator strength is onorder of magnitude larger.

The excitation of an electron from the HOMO to thLUMO is dipole forbidden in Zn2S2, as shown in Table I.The lowest-energy allowed excitation happens when an etron is excited from the 4b1u orbital to the LUMO, resultingin a 1B1u excited state. This transition absorbs 2.90 eV athe oscillator strength isf 50.015. The second allowed exc

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ELECTRONIC EXCITATION ENERGIES OF ZniSi CLUSTERS PHYSICAL REVIEW A 64 013201

tation occurs exciting an electron from the 3b3u orbital to theLUMO. This transition needs 2.97 eV andf 50.020, a bitlarger than the previous one. These two excitation enerlie within the range of the visible spectrum.

Two dipole-forbidden transitions have been reportedZn3S3. The only allowed transition calculated occurs excing an electron from the 6b1 orbital to the LUMO, yielding a1B1 state. The excitation energy is 4.32 eV, andf 50.038.This excitation energy is larger than the allowed excitatenergies of Zn2S2, and lies within the uv spectrum.

FIG. 1. Calculated global minima of ZniSi , i 52 –5, labeled,from left to right, GM(2), GM(3), and in the second row GM(4) andGM(5), respectively. Dark balls represent Zn atoms.

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The two calculated1A1g→1A2u transitions are allowedfor Zn4S4. The electron is excited from the 3a2u orbital tothe LUMO in one case, in which the excitation energy4.66 eV and the oscillator strength is 0.005. The other trsition is likely to be more intense, since the oscillatstrength is much larger, 0.088, and the excitation energonly 0.26 eV larger. In this last transition the electronexcited from the 3a2u orbital to the second virtual orbital8a1g .

Finally, three dipole-allowed transitions of Zn5S5 are col-lected in Table I. The first one occurs when an electron frthe HOMO is excited to the LUMO. The resulting excitestate has1A9 electronic structure. The excitation energy4.01 eV and the oscillator strength is 0.001. Exciting an eltron from the 35a8 orbital to the LUMO needs 4.21 eV, anthe oscillator strength is 0.001. The resulting excited sthas 1A8 electronic structure. Finally, the last calculated trasition is 1A8→1A9. Now, the electron is excited from thHOMO to the second virtual orbital, 37a8. The oscillatorstrength of this excitation is similar to the previous one0.001. The excitation energy is 4.39 eV, which lies in tultraviolet but close to the visible spectrum.

2. SKBJ(d) versus SKBJ(1sp2d2f) functionals

In Table 1, both the SKBJ(d) and SKBJ(1sp2d2 f ) exci-tation energies and oscillator strengths are given. Besithe differences between them may also be viewed thHaving a look at these differences, one may observethey are not very large. The largest difference in the exction energy is 0.07 eV for the1Ag→1B2g transition inZn2S2, which represents only 3% of the total excitation eergy. Nevertheless, most of the differences are smaller t0.04 eV, which is less than 1% of the total excitation en

TABLE II. SKBJ(d) excitation energies~eV! and oscillator strengths of small ZniSi , i 56 –9.

Transition SKBJ(d)Val. Conf. Orbitals States DE f

Zn6S6 21ag2 16bg

2 22bu2 22ag

2 17bg2 17bg→33ag

1Ag→1Bg 3.37 forbidden

LUMO:33ag 22ag→33ag1Ag→1Ag 3.37 forbidden

22bu→33ag1Ag→1Bu 3.60 0.029

Zn7S7 38a92 39a92 50a82 51a82 40a92 40a9→52a8 1A8→1A9 3.10 0.0001LUMO:52a8 40a9→53a8 1A8→1A9 3.27 0.0001

51a8→52a8 1A8→1A8 3.53 0.004

Zn8S8 50b2 51a2 52a2 51b2 52b2 52b→53a 1A→1B 3.65 0.0093LUMO:53a 51b→53a 1A→1B 3.65 0.0093

51a→53a 1A→1A 3.94 forbidden

Zn9S9 50a92 51a92 64a82 65a82 66a82 66a8→67a8 1A8→1A8 3.77 forbiddenLUMO:67a8 65a8→67a8 1A8→1A8 4.05 0.023

64a8→67a8 1A8→1A8 4.05 0.02362a8→67a8 1A8→1A8 4.40 0.03461a8→67a8 1A8→1A8 4.40 0.034

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MATXAIN, IRIGORAS, FOWLER, AND UGALDE PHYSICAL REVIEW A 64 013201

gies. Therefore, the SKBJ(d) results for the clusters of grou2 are expected to be reasonably accurate.

3. Group 2

Table II gives the SKBJ(d) excitation energies and oscilator strengths for the clusters of group 2, namely ZniSi , i56 –9, which are depicted in Fig. 2.

In Table II, three excitations are shown for Zn6S6. Two ofthem are dipole-forbidden, and the only one predicted todipole-allowed occurs from the ground state to the1Bu ex-cited state, where an electron from the inner 22bu orbital isexcited to the LUMO. The excitation energy of 3.60 eV linear the visible spectrum, and the oscillator strength is 0.0

For Zn7S7, three transitions are given in Table II. The twtransitions with smallest excitation energies occur from1A8 ground state to1A9 excited states. In one case, an eletron from the HOMO is excited to the LUMO, the excitatioenergy being 3.10 eV, and in the other case the electroexcited to the second virtual orbital, 53a8. The excitationenergy in this case is 3.27 V. In both cases, the oscillastrength is very small, 0.0001. The transition with largercillator strength, 0.004, occurs exciting an electron from51a8 orbital to the LUMO. The excitation energy of thitransition is 3.53 eV.

Zn8S8 has two dipole-allowed transitions and one dipoforbidden transition. The two allowed ones end up in1Bexcited states, and are near degenerate. The excitation, tfore, consists of promoting one electron from either the nedegenerate 52b or the 51b orbital into the LUMO. Its exci-tation energy is 3.65 eV and the oscillator strength is 0.0Exciting an electron from the inner 51a orbital to theLUMO, another excitation is obtained that ends up in the1A

FIG. 2. Global minima of ZniSi , i 56 –9, labeled, from left toright, GM(6), GM(7), and in the second row GM(8) and GM(9),respectively.

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excited state. Its energy is 3.94 eV, but the transition isdipole-allowed.

Finally, for Zn9S9, two groups of degenerate excitationwith large oscillator strengths have been predicted. In ogroup, the excitation energies are 4.05 eV and the oscillstrength is 0.023. These transitions occur when an electroexcited from the 65a8 orbital and from the 64a8 orbital tothe LUMO, respectively. In the other group, the excitatienergies are 4.40 eV, electrons being excited from the 6a8orbital and from the 61a8 orbital to the LUMO, respectively.The oscillator strength of both transitions is 0.034. Whenelectron is excited from the HOMO to the LUMO, the exctation energy is 3.77 eV, but the oscillator strength is zeAll five excitations happen from the1A8 ground state tovarious 1A8 excited states.

4. Group 1 versus group 2

The excitation energies of the clusters of group 1larger than those of group 2. The basic main structuralference between groups 1 and 2 is that structures in grouare rings, whereas the structures of group 2 are thdimensional spheroids. However, in all cases the excitatihappen from orbitals that are nonbondingp-type orbitals ofsulfur. This agrees with the bulk’s absorption behavior,its highest occupied band is thep band associated to sulfuatoms.

From a photovoltaic viewpoint, the excitation energiesspheroid clusters are the most interesting ones, since thecloser to the visible spectrum. The ring structures have etation energies larger than 4 eV, excepti 51,2, which makethem not very promising for photovoltaic applications. Tsmallest excitation energies of the spheroid clusters are lothan 4 eV. This is an interesting feature since the use of thspheroidal clusters in some hypothetical cluster-based somight increase the efficiency of the cell. We call for addtional experimental investigation of the optical-absorptiproperties of these clusters.

B. Excitation energies calculated as Kohn-Sham eigenvaluedifferences

The problem of the reliability of virtual orbitals withinDFT is a topic of great controversy@48#. Are the energies ofthese virtual orbitals reliable enough to calculate excitatenergies via Koopman’s theorem? It has been recently donstrated that for the exact exchange-correlation functioth excitation energies calculated as Kohn-Sham orbitalergy differences are very reliable@49#. Nevertheless, in reaproblems one does not use exact but approximate exchacorrelation functionals. In this section, we estimate the extation energies using the Kohn-Sham orbital eigenvaluwith B3LYP and MPW1PW91 approximate exchange corlation functionals. These functionals were chosen becaustheir great usability. In particular, the B3LYP functionalbecoming a standard for geometry optimizations andMPW1PW91 functional is one of the most employed aproximate exchange-correlation functionals for the calcution of electronic excitation energies. Results are shownTable III.

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ELECTRONIC EXCITATION ENERGIES OF ZniSi CLUSTERS PHYSICAL REVIEW A 64 013201

TABLE III. Excitation energies calculated with TDDFT@DE(1)#, MPW1PW91 Kohn-Sham HOMO-LUMO differences@DE(2)#, and B3LYP Kohn-Sham HOMO-LUMO differences@DE(3)#. D1 and D2denote the differences between MPW1PW91 and B3LYP with respect to TDDFT, respectively. All enare in eV.

Excitation DE(1) DE(2) D1 @DE(2)2DE(1)# DE(3) D2 @DE(3)2DE(1)#

Zn1S11S→1P 0.81 2.29 1.48~183%! 1.82 1.01~126%!1S→1S 3.96 4.75 0.79~20%! 4.36 0.40~10%!

Zn2S21Ag→1B2g 2.13 3.37 1.24~58%! 2.78 0.65~31%!1Ag→1B1u 2.90 3.95 1.05~36%! 3.40 0.50~17%!1Ag→1B3u 2.97 4.20 1.23~41%! 3.61 0.64~16%!

Zn3S31A1→1B1 3.60 4.75 1.15~32%! 4.21 0.61~17%!1A1→1A2 3.60 4.75 1.15~32%! 4.21 0.61~17%!1A1→1B1 4.32 5.49 1.17~27%! 4.94 0.62~14%!

Zn4S41A1g→1B1 3.74 4.88 1.14~30%! 4.48 0.74~20%!1A1g→1A2u 4.66 5.83 1.17~25%! 5.42 0.76~16%!1A1g→1A2u 4.92 6.01 1.09~22%! 5.48 0.56~11%!

Zn5S51A8→1A9 4.01 5.07 1.06~27%! 4.59 0.58~15%!1A8→1A8 4.21 5.26 1.05~25%! 4.78 0.57~13%!1A8→1A9 4.39 5.49 1.10~25%! 5.01 0.62~14%!

Zn6S61Ag→1Bg 3.37 4.33 0.96~29%! 3.87 0.50~15%!1Ag→1Ag 3.37 4.34 0.97~29%! 3.87 0.50~15%!1Ag→1Bu 3.60 4.51 0.91~25%! 4.04 0.44~12%!

Zn7S71A8→1A9 3.10 3.85 0.75~24%! 3.34 0.24~8%!1A8→1A9 3.27 4.29 1.02~31%! 3.96 0.69~21%!1A8→1A8 3.53 4.30 0.77~22%! 3.79 0.26~7%!

Zn8S81A→1B 3.66 4.56 0.90~25%! 4.04 0.38~10%!1A→1B 3.65 4.56 0.91~25%! 4.04 0.39~10%!1A→1A 3.94 4.82 0.88~22%! 4.35 0.41~10%!

Zn9S91A8→1A8 3.77 4.65 0.88~23%! 4.13 0.36~10%!1A8→1A8 4.05 4.90 0.85~21%! 4.38 0.33~8%!1A8→1A8 4.40 5.26 0.86~20%! 4.72 0.32~9%!

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A quick glance at Table III reveals that B3LYP orbitenergy differences~D2 column! performed remarkably bettethan MPW1PWP91 differences~D1 column!, as compared tothe TDDFT excitation energies. Also, it is observed thatthe cluster size increases, both orbital energy differencescloser to their corresponding TDDFT-predicted excitatienergy values. However, notice that the B3LYP orbital eergy differences do it faster.

Further inspection of Table III shows that the B3LYorbital energy differences for ringlike~group 1! clusters arealways greater that 1 eV. However, for the 3D spheroi~group 2! clusters, these differences are always less thaeV. Notice that for the largest cluster studied, Zn9S9, the

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deviation of the orbital energy differences with respect toTDDFT excitation energy is less than 0.4 eV, which repsents less than 10% of the total excitation energy. Hence,concluded that the B3LYP orbital energy differences pform reasonably well to estimate the excitation energies frthe Koopman theorem.

IV. CONCLUSIONS

TDDFT calculations yield interesting results for the exctation energies of ZniSi clusters. Ring like structure clusteri 52 –5, have larger excitation energies than thredimensional spheroidal clusters,i>6. However, for both

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MATXAIN, IRIGORAS, FOWLER, AND UGALDE PHYSICAL REVIEW A 64 013201

types of clusters, the excitation occurs from nonbondp-type orbitals of sulfur. Recall that for the two crystal strutures of bulk ZnS, wurtzite, and zinc blende, the highoccupied band is also associated with the nonbondingp-typeorbitals of the sulfur atoms.

The predicted electronic excitation energies of the sproidal clusters lie near the range of the visible spectrum,in some cases they are smaller than the bulk one. Therethose clusters constitute promising materials for use in ptovoltaic solar cells.

The difference between the B3LYP Kohn-Sham energof the orbitals involved in the sought after electronic excition appears to be a reliable practical approach to the ca

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lation of the electronic excitation energies for larger clustewhere TDDFT calculations become prohibitive. This aproach has been found to yield better results as the clusize increases. Nevertheless, its most salient drawback isno oscillator strengths are obtained.

ACKNOWLEDGMENTS

This research was funded by Euskal Herriko Uniberttatea ~the University of the Basque Country!, GipuzkoakoForu Aldundia ~the Provincial Government of Gipuzkoa!,and Eusko Jaurlaritza~the Basque Government!.

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