Electronic band structure of crystalline NaF: ionization threshold and excited states related to lattice defects

Surface Science 513 (2002) 127–139 www.elsevier.com/locate/susc

Electronic band structure of crystalline NaF: ionization threshold and excited states related to lattice defects Yuko Wasada-Tsutsui 1, Hiroshi Tatewaki

*

Institute of Natural Sciences and Computational Center, Nagoya City University, Mizuho-ku, Nagoya 467-8501, Japan Received 20 December 2001; accepted for publication 27 March 2002

Abstract We study the electronic band structure of crystalline NaF using the perfect and imperfect lattice cluster models embedded in the ionic cage with restricted Hartree–Fock calculations. The calculated ionization threshold is 7.9 eV, the valence bandwidth is 8.7 eV, the band gap is 12.2 eV, and the exciton band is 11.7 eV. The corresponding experimental values are 7.6, 8.3, 11.5–11.7, and 11.0 eV. All the experimental results are well explained by the present calculation. Consideration of lattice defects is vital in discussing the ionization properties, though the perfect lattice remains useful for treating properties of excited band states. The difference in the band parameter between NaF and LiF is also discussed and is explained by the difference of the Madelung potential working on F. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Ab initio quantum chemical methods and calculations; Electron density, excitation spectra calculations; Photoelectron emission; Alkali halides; Surface defects

1. Introduction Crystalline properties arise from both the periodicity of an ideal crystal and from imperfections due to lattice defects or impurities. The periodicity is responsible for the electronic energy band structure characterized by various band parameters, for example, while imperfections give rise to color variations and to ionic conductivity [1,2]. Alkali halides are quite in nature; they are colorless crys* Corresponding author. Tel.: +81-52-872-5791; fax: +81-52872-5791/3495. E-mail address: [email protected] (H. Tatewaki). 1 Research Fellow of the Japan Society for the Promotion of Science.

talline solids [3] and their structures have been well known from the early years of X-ray crystallography. They are also notable for spectroscopic phenomena such as the color-center [3] and excitons at relatively high temperatures [1,4]. The ionization threshold and the band gap are two of the most important parameters that characterize the electronic band structures of solids. The ionization threshold or work function gives an energetic information about the top of the valence band relative to the vacuum level [1,5]. The band gap that determines the electric conductivity is the energy difference between the bottom of the conduction band and the top of the valence band [1,5]. Systematic ultraviolet photoelectron spectroscopic (UPS) study of the alkali halides by Poole et al. [5,6] found the electronic structures of the valence

0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 1 6 9 9 - 0

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Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

bands, and gave the ionization thresholds, the bandwidths, and the band gaps. Systematic electron-energy-loss spectroscopy (EELS) studies of the alkali halides by Roy et al. [4] revealed the electronic structures based on color-centers and exciton bands as well as band gaps. The experimental ionization threshold determined by Poole et al. [5] is not really the top of the valence band because it is the mean of the valence band Ebexp ðF Þ relative to the vacuum; consequently the ionization threshold It is expressed as It ¼ Ebexp ðF Þ  0:5Etw ðF Þ 

ð1Þ

where Etw ðF Þ is the total width of the valence F2p band. The resulting value of It for crystalline NaF is 9.5 eV [5]. Based on the ionization spectrum given by Poole et al., the maximum of the UPS spectra is around 12.2 eV, and the spectra begin around 7.6 eV. Since the largest contribution to the Madelung potential for F ions comes from the closest Naþ ions, it is natural to suppose that lattice deficiencies on the Naþ ions have a strong influence on the electronic structure of F ions, leading to a smaller ionization threshold It than the value (9.5 eV) given in Ref. [5]. Many theoretical investigations exist of the electronic band structures of crystalline alkali halides using band theory and ab initio molecular orbital methods [7–12]. A cluster model considers a cluster embedded in the ion cage, with the interaction between the electrons of the cluster treated explicitly, and with the interaction between the electrons of the cluster and the rest of the crystal replaced with either point charges or appropriate ionic potentials at the respective ions. The cluster model is useful for treating imperfections in crystal structure. Its tractability depends on the size of the cluster and the assumed crystal field as well as the method used to treat the electronic structure, including electronic correlations. Tatewaki et al. [13,14] used a cluster model to study the electronic band structures of LiF by restricted Hartree–Fock (RHF) [15] calculations, where the cluster component is a unit cell of LiF (Liþ 14 F ). 13 In the present paper, we study the entire electronic band structure of crystalline NaF, including the ionization threshold, band gap, exciton band,

and valence bandwidth. RHF wavefunctions of a cluster model are used in the same manner as LiF [13,14] to approximate the electronic states of crystalline NaF. The difference in electronic states between the perfect and imperfect lattices is investigated to clarify the effect of the lattice defect on the electronic band structure, and it would be shown that the inclusion of the lattice defect is indispensable for discussing the ionization threshold. The electronic band structure of NaF will be compared with that of LiF in order to clarify the effect of the alkali ion on the electronic structure of alkali fluorides. Computational details are set out in Section 2. The band structure of NaF with respect to the perfect and imperfect lattice model is discussed in Section 3. Conclusions are given in Section 4.

2. Computational details 2.1. Cluster model for crystalline NaF We use a cluster model in which the cluster  Naþ 14 F13 is embedded in the center of the ionic cage of positive and negative point charges that represent Naþ and F ions in crystalline NaF. The embedded cluster is shown in Fig. 1. Imperfection  in the lattice is modeled by embedding Naþ 13 F13 , þ which has a defect at 11th Na adjacent to the

Fig. 1. Structure of the cluster embedded in the perfect lattice  (Naþ 14 F13 ). The lattice defect has been introduced by removing  Naþ at 11 (Naþ 13 F13 ).

Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

central F. The nearest neighbor Na–F bond distance adopted is that of the crystalline NaF, 4.3787 a.u. (see Fig. 1 and Ref. [16]). In the present NaF calculation, an ionic cage of 7  7  7 unit cells is adopted containing 3375 point charges. This gives a Madelung potential of 0.39911 a.u. [17] with error 6 105 (accurate value 0.39910 a.u.); the Madelung potential was evaluated by Evjen’s method [18].

Table 1 Basis set (5321/51) for Na and (52111/41) for F Atom

CGTF

Exponent

Coefficient

Na

1s

4096.9635259 616.3065206 139.9234486 39.0616210 11.9253590 20.6741577 1.9806481 0.6451695 0.5101884 0.0573054 0.0228090 75.3974467 17.2737343 5.1838656 1.6599580 0.5123127 0.061

0.0058321 0.0434867 0.1935901 0.4849272 0.4169838 0.0864294 0.5677187 0.5136044 0.1150463 0.6510125 1.0000000 0.0154377 0.0997514 0.3121288 0.4930726 0.3240055 1.000

2681.2646722 403.3656569 91.5392052 25.4776097 7.7609957 13.5300907 1.1814073 0.3277955 0.0369238 0.0129036 35.5035021 7.9530270 2.2960331 0.6732979 0.1683999

0.0059638 0.0443924 0.1964000 0.4873056 0.4119163 0.0804735 0.5811555 1.000000 1.000000 1.000000 0.0211069 0.1259955 0.3451725 0.4743937 1.000000

2s

3s1

2.2. Electronic structure calculations Using the RHF method, we have investigated the electronic structure of crystalline NaF under certain symmetry constraints. The symmetries of the clusters modeling the perfect and imperfect lattices are respectively D4h and Cs . The basis sets used for the electronic structure calculations are shown in Table 1. We construct the basis set for F as follows: exponents and contraction coefficients of the (53/5) basis set for the F atom [19] are reoptimized for F , and the valence s- and p-type contracted Gaussian-type functions (CGTFs) are split into two parts (521/41). Two diffuse s-type CGTFs are added to F in order to describe the transition corresponding to F2p ! F3s, where the two diffuse s functions are optimized in ð2p4 Þð3s1 Þ 4 P. The final CGTF is (52111/41). The valence stype CGTF of the (533/5) basis set for Na [20] is split into two, and a p-type polarization function [21] is also added. The resulting set for Na is (5321/51). We use the DSCF method for calculating ionization energies (IEs) and excitation energies (EEs): IEðDSCFÞ ¼ TEðRHF; ionized stateÞ  TEðRHF; ground stateÞ

129

3s2 2p

3p F

1s

2s1 2s2 3s1 3s2 2p1

2p2

Electron density maps are drawn by MOPLOT [24] and MOVIEW [25].

3. Band structure for NaF ð2Þ 3.1. Ionization threshold of the perfect lattice

and EEðDSCFÞ ¼ TEðRHF; excited stateÞ  TEðRHF; ground stateÞ;

ð3Þ

where TE represents the total energy. We use JAMOL 4 [22] as modified by Tatewaki and Miyoshi [13] for the cluster calculations, and the CGTF [23] program for basis set optimization.

The electronic configuration of the ground state  for the Naþ 14 F13 cluster embedded in the ion cage is ð   19a21g Þð   4a22g Þð   9b21g Þð   9b22g Þ  ð   13e4g Þð   2a21u Þð   14a22u Þð   6b21u Þ  ð   6b22u Þð   20e4u Þ

ð4Þ

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Table 2  The total energies, ionization energies, and GAOPs given by the perfect lattice (Naþ 14 F13 in D4h ) State

Ionized orbital

Ionization energy (eV)

Total energy (a.u.)

DSCF

Correlateda

11.67 11.69 14.29

13.67 13.69 16.29

b

Total energies and ionization energies 1 A1g (Gr. st.) 2 A2u 2 Eu 2 A2g

14a2u 20eu 4a2g

3533.552459 3533.123585 3533.122715 3533.027357

Ion F Positionc :

1st&3rd axis

Gross atomic orbital State 1 A1g (Gr. st.) 2 A2u 2 Eu 2 A2g

populations 10.077 10.068 10.068 10.073

Naþ 2nd corner

2nd center

1st&3rd center

1st&3rd corner

2nd axis

10.077 10.068 10.068 9.839

9.969 8.810 8.810 9.920

9.936 10.006 9.989 9.936

9.936 9.925 9.925 9.937

9.936 9.990 9.998 9.943

a

Values corrected by adding the electron correlation energy 2.00 eV. The experimental ionization energy is 9.5 eV [4]. c The 1st, 2nd, and 3rd represent the layers of the cluster. The 1st and 3rd layers have the same GAOPs by the symmetry constraint. b

Total energies and gross atomic populations (GAOPs) of the ground state and the three lowest ionized states are shown in Table 2. Ionization of an electron from 14a2u and 20eu gives the ionized states 2 A2u and 2 Eu respectively. The value of It calculated by DSCF is 11.7 eV, which is 2 eV larger than the experimental value. From the GAOPs given in Table 2 we see that electron ionization  occurs at the central F of Naþ 14 F13 . For the state that gives It , ionization is atomic-like, since the reduction in electrons at the central F is around 1.16 after ionization, and the other ions hold almost the same GAOPs as in the ground state. We know that the absolute value of the correlation energy for F is greater than that of F, suggesting that the ionization calculated by DSCF is less than the accurate ionization threshold of the crystal. From the experimental electron affinity (EA) of 3.4 eV [26] for gaseous F and the calculated SCF electron affinity 1.4 eV from the present basis set, we estimate that the correlation correction(DEcorr ) is 2.0 eV: EAðF : exptl:Þ ffi EAðF : DSCFÞ þ DEcorr DEcorr ¼ Ecorr ðFÞ  Ecorr ðF Þ

ð5Þ

We add this atomic DEcorr to the cluster IE(DSCF) and EE(DSCF) and henceforth give only the correlated results. The correlation correction increases the discrepancy between the calculation and experiment, and the correlated IE value of 13.7 eV is far from the experimental It ¼ 9:5 eV [5]. However, the present IE coincide with the results of the band calculation, 12.2 eV [8] and 14.7 eV [9]. We explain the discrepancy later. The hole distribution of 2 A2u given by a differential density between 2 A2u and 1 A1g , qð2 A2u Þ  qð1 A1g Þ, on the second layers is shown in Fig. 2. Fig. 2 shows that detachment of an electron creates a hole around the central F on the second layer. The 2 A2g lying 2.6 eV above the first ionized state is the state with a hole delocalized among the F atoms on the second layer (see Table 2). We find that the holelocalized states are always more stable than the hole-delocalized states. The wavefunction for the lowest ionized state is described by U ¼ CA j/p ðA1 BCD   Þj þ CB j/p ðAB1 CD   Þj þ CC j/p ðABC1 D   Þj þ   

ð6Þ

where /p represents all the p electrons. The first term implies that the p electron of an ion F A

Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

Fig. 2. Shape of the hole for the perfect lattice described as the differential density between the ionized state 2 A2u and the ground state 1 A1g , i.e. qð2 A2u Þ  qð1 A1g Þ on the second layer. The electron is largely removed from the central F.

is ionized, and the other terms have similar meaning. In closing this section, we emphasize that the IE calculated (13.7 eV) from the perfect lattice is far from the experimental value (9.5 eV). Further discussion will be given in Section 3.3. 3.2. Exciton and conduction band of the perfect lattice The excitation energies and GAOPs for the 16 lowest excited states are shown in Table 3. The lowest singlet and triplet states, lying 11.8 and 11.7 eV above the ground state, result from excitations of a 14a2u or a 20eu electron into 20a1g orbital. The excited states lying 0.4 eV above the lowest excited state arise from excitation from orbital 14a2u to 15a2u or from 20eu to 21eu . Using GAOP and electron density, we now argue that the lowest 3 A2u state is an excitonic state, and that 3 A1g , which lies 0.4 eV above the lowest 3 A2u is a conduction-band-like state. We expect that the lowest 3 A2u GAOP of the central F is around 9.0– 10.0, since single electron excitations are considered. However, the GAOP of the central F is negative, at )0.35. A similar tendency is found for the further three states lying 11.7–11.8 eV above the ground state. The remaining 10 states have normal GAOPs, although we list only five of these. To analyze the abnormal behavior of the GAOP of the central F, we separated the populations into

131

ion-core and excited electron GAOPs, which are shown in Table 4, together with 3 A1g and 3 Eg GAOPs. We see that the ion core GAOPs of the excited states and the ionized state are almost the same. This indicates that the large negative GAOP for 3 A2u and 1 A2u (9) arises from 20a1g , in which the excited electron moves. The negative GAOPs arises from large coefficients for the diffuse atomic orbitals in 20a1g (see Appendix A). The distribution of excited electrons given by the differential density 2 between 3 A2u and 2 A2u , qð3 A2u Þ  qð A2u Þ is shown in Fig. 3(a) with bird’s-eye views, and also in Fig. 3(b) with an isosurface of 0:0002 a:u:3 . The differential density over the space is normalized and integrates to unity. The excited electron is distributed interatomic region of the central and the remaining atoms. It is also distributed outside the twelve surrounding Fs, consistent with the molecular orbital coefficient of 20a1g . We recall that the hole is concentrated at the central F 2pz (see Fig. 2) and we have shown in Fig. 3(a) and (b) that the particle is strongly distributed to enclose the hole, although a little decrease is observed around Naþ . Further discussion on 20a1g is given in Appendix A. We recognize this as the excitonic excitation. The calculated excitation energy of 11.7 eV agrees with the experimental value of 11.0 eV. We next discuss excitations which involve the conduction-band-like states. The 3 A1g ð14a2u ! 15a2u Þ lies 0.4 eV above the excitonic excited states (see Table 3). The corresponding population analysis is set out in Table 3 as well as Table 4. The density of the particle given by qð3 A1g Þ  qð2 A2u Þ, is shown in Fig. 4. The GAOPs for 15a2u are large positive for F on the first and third layer, though they are small negative for Naþ on these. Fig. 4 shows that the excited electron is delocalized over the first and third layers. The flat distribution of the particle suggests that the 3 A1g state corresponds to excitation to a conduction band. The excitation energy of 3 A1g 12.2 eV agrees with the experimental band gap of 11.5 and 11.7 eV. The experimental energy difference between the band gap and the exciton band, 0.5–0.7 eV is therefore predicted accurately by the present study (0.4 eV). In summary, the perfect lattice model accurately describes the observed band gap and the observed exciton band. On the other hand, the

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Table 3  The excitation energies and GAOPs given by the perfect lattice (Naþ 14 F13 in D4h ) Electron removed orbital

Excited orbital

Singlet

Triplet

State (symmetry)

Ground state Exciton band-like stateb 14a2u 20a1g 20a1g 20eu Conduction band-like statec 14a2u 15a2u 20eux 21eux 20eux 21euy 21eu 14a2u 20eu 15a2u 14a2u 10b1g

1

A1g

1

A2u Eu

1

Excitation energya (eV)

State (symmetry)

Excitation energya (eV)

11.76 11.79

3

11.72 11.75

3

1

3

1

3

A1g A1g þ 1 B1g 1 A2g þ 1 B2g 1 Eg 1 Eg 1 B2u

A2u Eu

A1g A1g þ 3 B1g 3 A2g þ 3 B2g 3 Eg 3 Eg 3 B2u

12.17 12.17 12.19 12.52

12.15 12.15 12.17 12.17 12.19 12.52

Ion F d

Position :

1st&3rd axis

Naþ 2nd corner

Gross atomic orbital populations Exciton band-like state 3 A2u 10.378 1 A2u 10.420 3 Eu 10.378 1 Eu 10.419

10.378 10.413 10.378 10.422

Conduction band-like 3 A1g 3 Eg (a2u ! eu Þ 1 Eg ða2u ! eu Þ 3 Eg (eu ! a2u Þ 1 Eg ðeu ! a2u Þ

10.069 10.325 10.325 10.069 10.069

state 10.324 10.199 10.199 10.326 10.326

2nd center

1st&3rd center

1st&3rd corner

2nd axis

0.352 0.652 0.365 0.671

11.067 11.013 11.003 10.972

9.959 9.965 9.959 9.965

11.003 10.973 11.038 10.993

8.809 8.809 8.808 8.809 8.808

9.849 10.012 10.012 9.819 9.819

9.831 9.832 9.832 9.832 9.832

9.995 9.905 9.904 10.004 10.004

a The total energy for the ground state is 3533:552459 a.u. The RHF excitation energy is corrected by adding the electron correlation energy of 2.00 eV. b The experimental excitation energy of the exciton band is 11.0 eV [5]. c The experimental band gap is 11.5 [5] and 11.7 eV [4]. d The 1st, 2nd, and 3rd represent the layers of the cluster. The 1st and 3rd layers have the same GAOPs by the symmetry constraint.

calculated ionization energy IE is 4.2 eV larger than the value of It found by experiment (Eq. (1)) and is 6.1 eV larger than the onset value of It [6]. In Section 3.3, we discuss this disagreement. 3.3. Ionization threshold by the imperfect lattice The UPS spectrum given by Poole et al. begins at 7.6 eV, has a maximum at 12.2 eV, gives It of 9.5 eV, and ends at 15.9 eV. The resulting valence bandwidth is 8.3 eV. Our correlated IE of 13.7 eV based on the perfect lattice is larger than the ex-

perimental It . Correlation correction does not remove the discrepancy. Since the electronic field of the Naþ cations stabilizes the electronic field at the F anions, a deficiency in Naþ cations induces instabilities at F anions, which may lead to smaller ionization energies for the F anion. We, therefore, used an embedded model that has a defect at the 11th Naþ . Our calculations were performed with Cs symmetry. The electron configuration of the  ground state for the Naþ 13 F13 is ð   77a02 Þð   53a002 Þ:

ð7Þ

Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

133

Table 4  Partial population analysis for excited states of Naþ 14 F13 Positiona :

Ion F

State Ionized state 2 A2u Exciton band-like state 3 A2u Ion core 20a1g 1 A2u Ion core 20a1g

1st&3rd axis

2nd corner

10.068

10.068

8.810

10.006

9.925

9.990

10.068 0.310

10.068 0.309

8.824 9.176

10.002 1.065

9.926 0.032

9.987 1.017

10.067 0.353

10.068 0.346

8.846 9.498

10.000 1.013

9.926 0.039

9.985 0.988

10.069 0.000

8.810 0.000

10.004 0.155

9.925 0.095

9.992 0.004

10.068 0.258

8.809 0.000

10.008 0.005

9.925 0.093

9.989 0.084

Conduction band-like state 3 A1g Ion core 10.067 15a2u 0.257 3 Eg ða2u ! eg Þ Ion core 10.068 21eu 0.131 a

Naþ 2nd center

1st&3rd center 1st&3rd corner

2nd axis

1st, 2nd, and 3rd represent the layers of the cluster part. The 1st and 3rd layers have the same GAOPs by the symmetry constraint.

Fig. 3. Shape of particle for the perfect lattice described as the differential density between 3 A2u and 2 A2u , i.e. qð3 A2u Þ–qð2 A2u Þ, (a) shows the differential density on the second layer, (b) shows isosurfaces of the differential density, restricted to the cube where x, y, and z run from 10:0 to 10.0 a.u.; the origin is at the central F. The isosurface for 0:0002 a.u.3 is shown in light gray, and the isosurface for þ0:0002 a.u.3 in dark gray. The particle density is cut off at the boundary. The particle is distributed around the hole shown in Fig. 2. The distribution of the particle leans slightly to the corner Naþ .

The total energies and the GAOP of the ground state and the lowest five ionized states are listed in Table 5, together with the Madelung potentials where calculated IEs with correlation corrections (2.0 eV) are also included. The first and second IEs

of 7.9 eV are 5.8 eV lower than that of the perfect lattice model, and agrees with the onset (7.6 eV) of experimental It . By examining the molecular orbital coefficients, we find that an electron becomes ionized from the 2px in F1 and 2py in F15 orbitals

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Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139 

Table 6 are specified with ðNa14 F13 ÞðNa04 F05 Þ2 . The smaller basis set gives almost the same IE and excitation energies as the larger set: the differences in IE and excitation energies are 6 0:02 eV. Let us discuss IE. In gaseous F IE is 3.37 eV while for F in the ionic cage it is 14.20 eV. Introduction of six Naþ ions instead of the point charges more stabilizes the ionized state than the ground state and IE becomes smaller (13.33 eV). The introduction of 13 F ions destabilizes the ionized state and IE becomes larger (13.77 eV). We may find only a little decrease/increase in IE as the numbers of Naþ =F increase. We note that þ  ðNa14 F13 Þ and ðNa14 F13 ÞðNa4 F5 Þ2 , where the number of the cations and anions balances, give almost the same IE. The IE already converges þ around ðNa14 F13 Þ . We may also find a good convergence in the exciton states. From the values þ given in the table, we infer that ðNa13 F14 Þ in ionic cage gives a good model to simulate the crystalline NaF. Fig. 4. The particle distribution for 3 A1g in the perfect lattice given by qð3 A1g Þ–qð2 A2u Þ. Isosurfaces of the differential density is restricted to the cube where x, y, and z run from 10:0 to 10.0 a.u.; the origin is at the central F. The isosurface for 0:0002 a.u.3 is shown in light gray, and the isosurface for þ0:0002 a.u.3 is in dark gray. The particle density expanding out of the cubic region is cut off. The particle is distributed out of the first and third layers.

for the first and second ionized states, respectively, which are oriented toward the defect. The third and fourth ionized states are produced by detachment of an electron from F1 and F15 2pz orbitals, respectively. We conclude that lattice defects imply smaller IE, explaining the onset of experimental It value. 3.4. Convergence on the cluster size Changing the cluster sizes, we calculated the first IE and the excitation energies of the exciton states. Results are collected in Table 6. For the cluster  þ ðNa14 F13 ÞðNa4 F5 Þ2 , ions added to ðNa14 F13 Þ are calculated by the two sets, one of which is that collected in Table 1 and the other minimum set plus one 3s cGTFs. The results given by the latter in

3.5. Comparison with others The ionization threshold, band gap, exciton band and valence-bandwidth calculated here are listed in Table 7, together with other theoretical and experimental results. The experimental ionization threshold It of 9.5 eV could be regarded as a lower limit of the onset of the perfect lattice model because it includes the influence of lattice defects. The ionization threshold of 13.7 eV for the perfect lattice model in this work is to be compared with the experimental value of 9.5 eV. The ionization threshold obtained via the top of the valence band ranges from 9.4 eV [11,12] to 14.7 eV [9]. Since these are also based on the perfect lattice, the results of the present perfect lattice model can be compared to these band values. The band gaps calculated by band theory range from 9.6 eV [10] to 13.4 eV [11], depending on the calculation method. In particular, the calculated band gaps, 11.4 eV [9], 12.0 eV [12], and 12.7 eV [8], agree with the experimental value of 11.5 or 11.7 eV, and with the present work 12.2 eV. The band gaps of a cluster model embedded in an ion cage calculated with unrestricted Hartree–Fock (UHF) and single-excitation configuration inter-

Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

135

Table 5  a The total energies, ionization energies and GAOPs given by the imperfect lattice model (Naþ 13 F13 in Cs ) State

Ionized orbital

Total energies and ionization energies 1 0 A (Ground state) 2 0 A 75a0 2 0 A 77a0 2 00 A 52a00 2 00 A 51a00 2 00 A 53a00 Ion

Positionc

Total energy (a.u.)

Ionization energyb (eV)

3372.706896 3372.491154 3372.491052 3372.483089 3372.480316 3372.426018 Numberd

7.87 7.87 8.09 8.17 9.64 Madelung potentiale (a.u.)

Gr. st.

1st 2 A0

2nd2 A0

0.1707 0.2970 0.2673 0.2673 0.1707 0.1707 0.1707 0.2970 0.2970

10.053 10.075 10.082 10.084 9.988 10.053 10.054 10.075 10.075

10.044 10.070 10.075 10.075 8.897 10.044 10.044 10.070 10.070

10.045 10.076 10.074 10.082 9.959 9.072 10.055 10.076 10.077

2 3 4 5 6 12 13 14

0:5606 0:5606 0:5606 0:4923 0:4923 0:5133 0:5606 0:5606

9.926 9.938 9.938 9.939 9.938 9.952 9.927 9.927

9.976 9.933 9.933 9.932 9.932 9.981 9.976 9.976

9.926 9.967 9.933 9.932 9.938 9.941 9.949 9.925

11

0:3991

Madelung potential and gross atomic orbital populations 1st&3rd 7 F 8 9 10 2nd 1 15 16 17 18 Naþ

1st&3rd

2nd

Defect

2nd

GAOP

a

The experimental onset of the ionization threshold is 7.6 eV. Values corrected by adding the electron correlation 2.00 eV. c The 1st, 2nd, and 3rd represent the layers of the cluster. The 1st and 3rd layers have the same GAOPs by the symmetry constraint. d See Fig. 1. e  The Madelung potential on F of the perfect lattice model (Naþ 14 F13 ) is 0.3991 a.u. b

action (CI-Singles) are 11.6 and 11.9 eV respectively [7]. These also agree with experiment, showing again that the embedded cluster model is suitable for investigating crystalline, NaF. The exciton band calculated by Kunz, 9.8 eV [11,12], is lower than the experimental value of 11.0 eV. By contrast the exciton band at 11.7 eV calculated here is in good agreement with the experimental value. The valence bandwidth of 2.7 eV calculated by Kunz [12] is similar to our value of 2.9 eV from the perfect lattice model. This shows that if the lattice were perfect, the experimental

total bandwidth would be around 3 eV. However, the theoretical values are close to the experimental half width 2.9 eV rather than the experimental total width Etw 4.9 eV [6]. We believe that the calculated bandwidth should not be compared with half width, but with the total width. We are skeptical of the experimental Etw 4.9 eV, since no attention was paid to the onset It value as discussed above. We quote Etw directly from UPS results given by Poole as 8.3 eV, which is near to the calculated value of 8.7 eV with a lattice defect; the onset It must be considered in order to evaluate

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Table 6 Convergence of the first ionized state and the exciton states versus the cluster size Cluster

First ionized statea

F in gaseous phase

2

Exciton statesa

A2u

3.37 eV

1

A2u A2u 1 A2u 3 A2u 3 A2u 1 A2u 1 A2u 3 A2u 1 A2u 3 A2u 1 A2u 3 A2u 1 A2u 3 A2u 1 A2u 3 A2u

4.91 4.89 8.88 8.64 10.88 10.83 10.93 10.89 11.76 11.72 11.93 11.89 11.94 11.91 11.98 11.94

3

F in i.c.b

2

A2u

14.20 eV

ðNa6 FÞ5þ in i.c.

2

A2u

13.33 eV

ðNa6 F13 Þ7 in i.c.

2

A2u

13.77 eV

ðNa14 F13 Þþ in i.c.

2

A2u

13.67 eV

(Na14 F13 )(Na4 F5 ) 2 in i.c.

2

A2u

13.66 eV

ðNa14 F13 ÞðNa04 F05 Þ 2 in i.c.

2

A2u

13.68 eV

in i.c. ðNa14 F13 ÞðNa04 F05 Þ2 F05 4

2

A2u

13.75 eV

a b

Values corrected by adding the electron correlation 2.00 eV. The symbol i.c. means ionic cage.

Table 7 Theoretical and experimental band structure of NaF (eV) Onset of It

Band gap

Valence bandwidth

Exciton band

Imperfect

Perfect

Perfect

Imperfect

Perfect

7.9

12.2 12.7d , 11.4e , 9.6f 13.4b , 12.0c 11.6, 11.9g 11.5h , 11.7i

2.9 2.7c

8.7

11.7 9.8b ;c

4.9h , 8.3j

11.0h

Theory

Ionization threshold ðIt Þa Perfect

This work Band theory

13.7 9.4b; c , 12.2d 14.7e >9.5h

7.6h;j

Cluster model Experimental a

The ionization thresholds given by band theories are the absolute values of the top of the valence band. See Ref. [11]. c See Ref. [12]. d See Ref. [8]. e See Ref. [9]. f See Ref. [10]. g See Ref. [7]. h See Ref. [6]. i See Ref. [5]. j The onset It was directly read from UPS spectrum given in Ref. [5]. b

the total bandwidth. The effect of the lattice defect is vital in evaluating the ionization threshold as well as the valence bandwidth. In summary, calculations of properties related to the ionization energy, such as the valence bandwidth and ionization threshold, cannot agree with experiment unless lattice defects are accounted for. By contrast, calculated properties related to excited states, such as the band gap and

bulk exciton, are accurately reproduced by the perfect lattice model. As a conclusion, experimental band parameters are almost thoroughly explained by the present calculations. 3.6. Comparison of crystalline NaF with LiF A summary of the electronic band structures of crystalline NaF and LiF is given in Table 8. The

Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

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Table 8 Electronic band structure of LiF and NaF (eV) Ionization threshold Perfect NaF LiFa Differenceb a b

Calc. Exp. Calc. Exp. Calc. Exp.

13.7 9.5 15.2 9.8 1.5 0.3

Imperfect 7.9 7.6 8.7 7.5 0.8 0.1

Band gap

Valence bandwidth

Perfect

Perfect

12.2 2.9 11.5, 11.7 4.9 13.9 2.7 13.6 6.1 1.7 0.2 2.1 1.2

Exciton band Imperfect

Perfect

8.7 8.3 9.8 10.5 1.1 2.2

11.7 11.0 13.4 13.5, 12.6 1.7 2.5

Distance (a.u.)

Madelung potential on F

4.379

10.9

3.796

12.5

0.583

1.7

See Table 6 in Ref. [14]. The difference in the band parameters between NaF and LiF.

theoretical ionization threshold of LiF based on the perfect lattice model is 1.5 eV larger than that of NaF. The calculated band gap and exciton band are also 1.7 eV larger than those of NaF. In both LiF and NaF the calculated differences run parallel to the experimental differences except for It , for which the lattice defect plays an important role. The Madelung potential on F in the perfect lattice model of LiF is 0.4604 a.u. and in NaF is 0.3991 a.u. The difference between these Madelung potentials of 0.0613 a.u. (1.7 eV) causes the differences in electronic band structures discussed above; the smaller Madelung potential in NaF results from the larger distance of the nearestneighbors (4.379 a.u.) than the distance between Li and F (3.796 a.u.). When ionization is considered, the electron to be ionized is confined to F by the Madelung potential, and the ionized electron is free from the Madelung potential. The ionization threshold It is therefore directly influenced by the Madelung potential on F . However, the electron excitation should be different from the electron ionization because the excited electron remains under the control of the Madelung potential. Ordinarily, (a) the difference of Madelung potential before and after the excitonic excitation should be zero because the Madelung potential is the same for F before and after the excitation; and (b) the difference of Madelung potential in the band gap calculation should be twice as large as the F Madelung potential because the valence band orbitals consist of s orbitals of the cations, con-

trolled by of the negative Madelung potential and suggesting the larger band gap difference (3.4 eV) between LiF and NaF [1]. However, only the Madelung potential difference (1.7 eV) on F affects the resulting band parameters. We discuss first (b) and next (a). Recall that in the valence band-like excited states, the electrons are uniformly delocalized on the first and third planes (see Fig. 4). As a result, the positive and negative Madelung potentials cancel in the excited state orbital, giving a band gap difference of 1.7 eV between LiF and

Fig. 5. The excited orbital 20a1g of the excitonic state of þ   Naþ 14 F13 . The amplitude of the orbital of Na14 F13 along the yaxis is shown. Suffixes attached to Na and F refer to the ion numbers given in Fig. 1. The 3s orbital ð2p4 Þð3s1 Þ 4 P of gaseous F is shown for comparison.

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Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

NaF. Let us now discuss the excitonic excitation (a). We saw in the previous section that, in the exciton state, the particle is heavily distributed around the central F and the surrounding 12 F . However, even the particles around the central F are sandwiched by Naþ , as shown in Fig. 3(a) and (b); we cannot therefore disregard the cation potential (See also the orbital amplitude given in Fig. 5). The difference in the calculated excitation energy to the exciton band between LiF and NaF is 1.7 eV, the cancellation of positive and negative Madelung potentials in the excitonic states of LiF and NaF being almost exact. In summary, the difference in excitation energy between LiF and NaF is determined predominantly by the difference in Madelung potential on F of these clusters.

that the lattice defect influences strongly the experimental ionization value. The difference in band parameters between NaF and LiF has also been discussed. This difference can be explained wholly on the basis of the magnitude of the Madelung potential.

Acknowledgements The calculations in this work were performed on the IBM SP computer of the Computer Center of Nagoya City University. The present research was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

Appendix A. The detailed analysis on 20a1g 4. Conclusions In this appendix, we analyze the shape of 20a1g . The MO was decomposed into three AO groups: (1) AOs from the central F , (2) AOs from the surrounding 12 F , and (3) AOs from the 14 Naþ . The amplitudes 20a1g and three components of Naþ 14 F13 are plotted along the y-axis (Fig. 5). In Table 9 we give the coefficients only for the inner 3s (3s1 ) and outer 3s (3s2 ) whose contributions to 20a1g are significant. Molecular orbital coefficients for the p orbitals on the central F vanish because of the symmetry. As expected from the large negative coefficients, the AOs from the central F give a very large negative amplitude around the central F , which is compensated by the AOs of group (3) and group (2). The resulting 20a1g has some similarities to 3s of 2p4 s4 P of the gaseous F. Since the contributions from group (2) change very slowly

We have studied the entire electronic band structure of crystalline NaF using the perfect and imperfect lattice cluster models embedded in the ionic cage with RHF calculations. Our calculated values of band parameters are in good agreement with the experimental values. Consideration of lattice defects is essential in deriving ionization properties, though the perfect lattice remains useful for treating excited states. For example, the IE calculated using the perfect and imperfect lattice are 13.7 and 7.9 eV respectively. The latter agrees closely with the onset of ionization threshold It (onset of UPS spectrum), at 7.6 eV [6]. The experimental ionization threshold is reported as 9.5 eV, which is far from the calculated value of 13.7 eV based on the perfect lattice model, indicating Table 9  MO coefficients for 20a1g of exciton state of Naþ 14 F13 Object position:

Object atom F

Na

1st&3rd axis Number of equivalent ions MO coefficients 3s2 3s1

8 0.491 0.064

2nd corner 4 0.496 0.074

2nd center 1 9.977 3.070

1st&3rd center 2 1.765 0.172

1st&3rd corner 8 0.171 0.191

2nd axis 4 1.700 0.157

Y. Wasada-Tsutsui, H. Tatewaki / Surface Science 513 (2002) 127–139

along the y-axis, the Naþ and the central F AOs are responsible for the shape of 20a1g .

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