Dipole surface plasmon in K+N clusters

Solid State Communications, Voi. 84, No. 9, pp. 905-909, 1992. Printed in Great Britain.

0038-1098/92 $5.00 + .00 Pergamon Press Ltd

DIPOLE S U R F A C E PLASMON IN K~ CLUSTERS F. Garcias Departament de Fisica, Universitat de les llles Balears, E-07071 Ciutat de Mallorca, Spain M. Barranco and M. Pi Departament E.C.M., Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain and J. Navarro Departament de Fisica At6mica i Nuclear and IFIC, Universitat de Val6ncia, Dr. Moliner 50, E-46100 Burjassot, Spain

(Received 19 June 1992 by F. Yndurain) The technique of sum rules has been used to investigate the dipole surface plasmon for K~ clusters within a Density Functional Theory and the spherical jellium model. The role played by non-local effects is discussed comparing the results obtained from different functionals. Band-structure and core-polarization effects have been phenomenoiogically included in the calculation by means of an electron effective mass and a dielectric constant. Comparison with recent experimental data is presented. THE PHOTOABSORPTION cross section of singly clusters. Specifically, we would like to address which ionized clusters of potassium has been recently possible mechanisms could explain, at least in part, measured by the Orsay group [!, 2]. Measurements the disagreement between theory and experiment. We use the RPA sum rules technique, an on the small clusters K+ and K~t were performed detecting the fragmentation produced by unimolecu- approach that has proved to be very useful in lar evaporation after single photon absorption [I]. A Nuclear Physics, in particular for the characterizanew procedure based on multistep photoabsorption tion of the so-called giant resonances [5, 6]. This is a experiments has permitted us to obtain the cross rather simple-to-use method that has also been section for the large clusters K~00 and K~ 0 [2]. All the employed to describe the response of near-free spectra are found to be dominated by a single electron metals to different external fields [7-12]. It collective state, which is well fitted by a Lorentzian is worth noting that for K~ clusters the surface from which the frequencies and widths have been plasmon is below the ionization threshold, and that determined. The dipole frequency wd for large spheres its strength is concentrated in the collective state. This was deduced in [2] from the measurements of the makes the use of sum rules especially well suited for surface [3] and bulk [25] plasmon dispersion relations, the present study. employing the classical expressions for these frequenSum rules are defined as cies. Thus, the variation of the plasmon frequency (1) with size could be analyzed. The same group also Mk = ~_, (f. - fo)kll2, performed some calculations of the strength within the Random Phase Approximation (RPA) in the where Q is an operator representing the external field Local Density Approximation (LDA), using the acting on the system (the dipole operator )-'~,-z,. in the numerical code of Bertsch [4] that they have adapted case we are interested in), and En are the eigenvalues for large clusters. The theoretical frequencies are blue of the eigenstates In). These sum rules are the energy shifted as compared with the experimental ones, by moments of the strength function typically 20%. Motivated by these measurements, we have S(E) : ~ 6(E, - E)I(OIQIn)I2. (2) n calculated the dipole surface plasmon in K~ 905

906

DIPOLE S U R F A C E PLASMON I N / ~

CLUSTERS

Vol. 84,'No. 9

Table 1. Total energy (E), rms radius and dipole E3 energy for the clusters K~93and K~'I39 obtained within different frameworks. KS: Kohn-Sham. Central columns." TFD W with different values of the coefficient ~. TF-h4. • Using the h4 order semiclassical expansion of the kinetic energy KS

fl = 1

/~/= 1/2

fl = I/9

TF-II 4

K~-93

E(au) r (au) E3 (eV)

-7.11 17.13 2.44

-6.45 17.23 2.41

-6.84 17.13 2.45

-7.24 17.00 2.49

-7.17 17.10 2.45

g~139

E(au) r (au) E3 (eV)

-10.77 19.54 2.46

-9.90 19.62 2.43

-10.40 19.55 2.45

-10.93 19.46 2.49

-10.85 19.53 2.46

Odd moments of S(E) can be obtained with RPA precision with the only knowledge of the K o h n Sham (KS) ground state (g.s.) [8-10]. The interest of these sum rules lies in that the average energy E and the variance t7 of the strength are bound as

Weizs/icker ( T F D W ) approximation to the kinetic energy density:

E, < E' _< E 3

because the numerical problem is then reduced to solving an Euler-Lagrange equation to determine the electron density. This approximation simplifies the calculations, especially for large systems, giving similar results to the KS ones, as can be seen in Table 1. There we display some results for the clusters K~3 and K~39 obtained using the Slater exchange term and a correlation energy of Wigner type as in [10]. The original Weizs~icker approximation corresponds to fi = 1, and the value B = 1/9 was deduced by Kirzhnits [13] through a semiclassical expansion of the kinetic energy density up to h 2 order. However, this value results in too steep densities as compared with the KS ones. Previous studies have led us to conclude that a value bigger than 1/9 should be used in the T F D W approximation. For instance, in [14] the value /~ = 1/5 was used to reproduce the g.s. energy of atoms, and in [15] the value/~ = 2/9 was employed to study the polarizabilities of small metal spheres. We take the heuristic value/3 = 1/2, already used in a calculation of the polarizability of metal clusters [16], because it yields results in good agreement with the KS ones. As a further test we have also employed the expression for the kinetic energy density given by a semidassical expansion up to h 4 order [l 7, 18], and have obtained the density in a fully variational calculation, as in [18]. As it can be seen in Table 1, the approximations KS, T F D W (fl = 1/2) and TF-h 4 provide results which are not significantly different, in particular for the E 3 energies. In the following, the results labelled T F D W will correspond to the value/~ = 1/2. The E 3 and El energies of some K~ clusters obtained in the T F D W approach are shown in

(3)

and l 2 - E 2), a 2 < ~(E~

(4)

where we have defined the mean energies

( Mk '~1/2 ek = \M----~k-~/ "

(5)

Consequently, one may estimate the centroid and the variance of the strength function by evaluating the three RPA sum rules M_t, M~ and M 3. If most of the strength is in a narrow energy region, then El and E 3 are good estimates of the mean energy of the resonance. We can take advantage of this fact and use E3 to represent the average energy. It is worthwhile stressing that in all approximations we will discuss below, the calculation of Mi and M 3 only requires the evaluation of one dimensional integrals involving g.s. quantities. This is not quite the case for M_ I [9-11]. The valence electron cloud is described by means of a density functional consisting of the kinetic energy density, the Coulomb direct plus exchange terms, and a correlation energy density. The neutralizing positive background is approximated by a spherical jellium. These ingredients are quite usual in this sort of calculations, and they provide reasonable g.s. results. After solving the KS equations, one calculates the sum rules, which only require the knowledge of the g.s. electron and kinetic energy densities. For large clusters, we have found convenient to use the T h o m a s - F e r m i - D i r a c -

r(r) = ~3(371.2)2/3n5/3 -I ~4 (Vn)2n

(6)

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DIPOLE SURFACE PLASMON I N / ~ CLUSTERS

907

Table 2. Dipole surface plasmon energies E 3 and El in eV for K~ clusters obtained with the indicated effective values of c and m*. Experimental results are from [I, 2] =1.07, m * = 1.25

e=l,m*=l TFDW

AG

WDA-PB

TFDW

AG

N

Exp.

E3

E1

E3

El

E3

El

E3

E1

E3

Et

9 21 93 500 900 1000 2000

1.93 1.98

2.39 2.40 2.45 2.48 2.49 2.49 2.50

2.33 2.33 2.37 2.43 2.44 2.45 2.46

2.32 2.36 2.42 2.45 2.48 2.48

2.16 2.21 2.28 2.35 2.36 2.39

2.22 2.28 2.32

2.13 2.16 2.27

2.14 2.15 2.18 2.20 2.21 2.21 2.22

2.10 2.11 2.13 2.17 2.18 2.18 2.19

2.08 2.10 2.15 2.19 2.20

1.95 1.99 2.04 2.10 2.13

tod

2.03 2.05

2.24

2.54

Table 2 under the heading e = 1, m* = 1. As it could be expected, these energies are about 20% higher than the experimental ones and slowly tend to the asymptotic value top/re3, where top is the bulk plasmon frequency. We use rs = 4.86 atomic units (a.u.) for potassium. One may wonder to what extent non-local effects not considered in the previous density functional could modify these results. Indeed, a wellknown drawback of the LDA is that the exchangecorrelation potential decays exponentially at large distances. If the potential had the correct - 1 / r behavior, the electronic spill-out would increase, and consequently the plasmon frequency would decrease. To go beyond the LDA we have used the semiclassical functional derived by Alonso and Girifalco (AG) [19], which includes non-local effects. They come into play through a weighted density, determined by the requirement that the normalization of the one-electron exchange hole be fulfilled at each point. The resulting exchange plus correlation potential behaves asymptotically as -1/2r. The corresponding E3 energies are displayed in Table 2 (label AG, ~ = 1, m* = 1). Although they have decreased with respect to the TFDW ones, the changes are small and in practice only appreciable for small clusters. The calculation of M_t (thus El) in the AG approximation demands the solution of a rather involved integro-differential equation. Since E3 suffices to give the overall N-dependence of the plasmon energy, what we call E1 in this approximation actually is the result of solving the much simpler TF integro-differential equation [16] using as an input the AG g.s. density. One can see that this hybrid El energy very much behaves as E3. A more elaborate way to include non-local effects

2.25 has been used in [20]. There, the correct asymptotic behavior is imposed by conveniently symmetrizing the exchange plus correlation potential. The price paid for this procedure is that the consistency between the potential and the density functional is lost. We display in Table 2 under the heading W D A PB, the corresponding E 3 and E I energies taken from [20]. A substantial decrease of the average energies is obtained for small clusters. However, we would like to point out that these non-local effects disappear in the bulk [19, 20] and consequently have little influence in large clusters. For this reason, we have stopped these calculations, which are technically quite involved, at N = 93. As a matter of fact, the above discrepancies between theoretical and experimental results are not new. They have been found, for instance, in calculations of cluster polarizabilities [9, 16, 21, 22] and bulk plasmon dispersion relation [23]. In this latter case, the origin of the disagreement has been attributed to atomic lattice effects [24-26], which are beyond the simple jellium model, and show up through band structure and polarization of core electrons. In the bulk metal, the positive background in which the valence electrons move is periodic and this effect can be incorporated through an effective mass m*, also called crystalline mass [27]. Replacing the bare electron mass by the effective one would take into account the non-uniformity of the ion distribution. On the other hand, the core electrons constitute a polarizable background, which in some cases can be characterized by a static dielectric constant e, It has been shown [23] that using m* and ~, a good agreement with experimental bulk plasmon dispersion data is obtained for high-density metals like Na and AI. This encourages one to further explore whether or

908

DIPOLE S U R F A C E P L A S M O N I N / ~

not it is possible to use the same heuristic values for these quantities to simultaneously describe the bulk and surface plasmon dispersion relations, as well as the dipole mode in clusters and the limit of a large sphere. Introduicng m* and e, the volume plasmon frequency reads (in a.u.) ~v2 = •p2 em* '

(7)

whereas for a plane surface we have

2_ ~p2 ~sp (~ + 1)m*

(8)

and for the frequency of the dipole mode of a large sphere (~ + 2)m*"

(9)

CLUSTERS

Vol. 84, No. 9

Altogether, we obtain an appreciable downward shift of the energies, which is practically independent of size. In the lightest cluster, the energies still lie some 10% above the experimental data. On the light of the preceding results, the following picture emerges. In light clusters, a combination of non-local effects in the density functional and atomic lattice effects can bring the theoretical results close to the experimental values. In large clusters, non-local effects have no practical influence on the dipole energy and an appreciable discrepancy still remains. However, due to the experimental technique employed in their study, large clusters have been analyzed in a rather hot state (around 600-800 K, see Ref. [2]). This makes room to attribute the remaining discrepancy to thermal effects. In the lack of any better prescription, one could consider the thermal dilatation of the jellium sphere by effectively increasing the bulk electron radius per electron rs. Using the experimental linear expansion coefficient of potassium [30] a = 8.3 x 10-5 K -I, we have

Neither m* nor e are experimentally well-known quantities. The dielectric constant e is not available for the metal state and it is usually obtained from rs(T ) ,,~ rs(TO)[1 + t ~ ( T - To)]. (10) the experimental ion polarizability [28] using the Clausius-Mossotti formula, or from the experimen- Taking To = 300K and rs(To) = 4.86a.u., equation tal atomic susceptibility [29]. The effective mass (10) yields rs(800 K) = 5.05 a.u., i.e., a 4% larger than should be determined from the study of intraband the value we have previously used. Employing transitions. We shall give heuristic values to m* and e rs = 5.05a.u., ~ = 1.07 and m * = 1.25, we get for by adjusting the bulk and the surface plasmon K~ 0 the values El = 2.06eV and E3 = 2.09eV in energies to their experimental values at q = 0. In reasonable agreement with the experiment. the case of potassium, these values are w(0) = 3.81 eV [24, 25], and ~sp(0)= 2.74eV [3]. Using the above Acknowledgements - We would like to thank Nuria formulae we get E = 1.07 and m * = 1.25, and the Barber~in and Lloren~; Serra for useful discussions. dipole mode is predicted at wd = 2.25 eV. This result We are very indebted to Angel Rubio for supplying is very similar to those deduced in [2], in spite that the us with the W D A - P B results corresponding to K~3. This work has been supported in part by the effect of ¢ and m* has not been explicitly considered DGICYT grants PB89-0332 and AEN90-0049. there. As mentioned before, the relations between classical frequencies Wd = V / ~ a;sp(0) and REFERENCES a;d = w(0)/V~ have been employed in this reference, obtaining 2.24 eV and 2.22 eV, respectively. Note that 1. C. Brrchignac, Ph. Cahuzac, F. Carlier & J. Leygnier, Chem. Phys. Lett. 164, 433 (1989). in this case one assumes that atomic lattice effects 2. C. Brrchignac, Ph. Cahuzac, N. Kebaiqi, modify both frequencies in the same way, contrarily J. Leygnier & A. Sarfati, Phys. Rev. Lett. 68, to what is indicated in equations (8) and (9). 3916 (1992). However, since the value obtained for ~ is close to 3. K.D. Tsuei, E.W. Plummer & P.J. Feibelman, unity, all these determinations of a;a are very similar. Phys. Rev. Lett. 63, 2256 (1989). In Table 2 we display the T F D W and AG results 4. G. Bertsch, Comput. Phys. Comm. 60, 247 obtained using the effective values e---1.07 and (1990). 5. O. Bohigas, A.M. Lane & J. Martorell, Phys. m * = 1.25. The assumption of a constant core Rep. 51, 267 (1979). polarization is valid when the characteristic fre6. E. Lipparini & S. Stringari, Phys. Rep. 175, 103 quency of core-valence electron transitions is large (1989). compared with the plasmon frequency, i.e., when 7. A.A. Lushnikov & A.J. Simonov, Z. Physik the core electrons are tightly bound, as is the case 270, 17 (1974). for alkali metals. The effective mass does vary with 8. G. Bertsch & W. Ekardt, Phys. Rev. B32, 7659 size, and the use of a constant value should be taken (1985). 9. M. Brack, Phys. Rev. B39, 3533 (1989). with some caution, especially for small clusters.

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DIPOLE SURFACE PLASMON I N / ~ CLUSTERS

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