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Unified model for the electronic and atomic shell structures of metallic clusters
NANoSTRUCTURED MATERIALS VOL. 3, PP. 301-309, 1993 COPYRIGHT©1993 PERGAMONPRESSLTD. ALL RIGHTSRESERVED.
0965-9773/93 $6.00 + .00 PRINTEDIN THE USA
UNIFIED MODEL FOR THE ELECTRONIC AND ATOMIC SHELL STRUCTURES OF METALLIC CLUSTERS
P. Stampfli and K.H. Bennemann Institute for Theoretical Physics, Freie Universit~t Berlin Arnimallee 14, W-1000 Berlin 33, Germany
Abstract--We present a theory explaining from a general point of view the intensity anomalies (magic numbers) in the mass spectra of metallic clusters. The transition from an electronic shell dominated structure in the mass spectra of smaller clusters to an atomic shell dominated structure in the mass spectra of larger clusters is discussed as a function of cluster size n and temperature T. We estimate where in the (T, n)plane electronic- and where atomic shell structure dominates. Results are given for the shell structure in the cohesive energy, the ioni:ation potential and photoyield, which are useful for distinguishing between the two kinds of structure. INTRODUCTION The mass spectra of metallic clusters have been for many years an interesting tool for learning about the electronic and atomic structure of metallic clusters (1). The mass-intensity spectrum observed for smaller clusters of many metal-atoms led to the invention of the giant atom model for clusters and the resultant electronic spectrum with characteristic electronic shell structure (2). The electronic energy spectrum was characterized by shells. These consist of several energy levels el, which are nearly degenerate although they have different angular momenta 1. Peaks in the mass spectrum resulted when the difference An = (En -En-1) between cluster energies En changed abruptly, as is the case when a shell is f'dled by electrons and the next higher energy levels are separated by a gap. Thus, the essential features of the mass spectra of many smaller metallic clusters could be explained (3). Recently, intensity anomalies (magic numbers) of larger clusters (s. Nan: 1500 < n < 22000) have been observed and interpreted as resulting from an atomic shell structure of the metallic clusters (1), Intensity maxima of ionized clusters resulting from photoionization have been observed at half-filled surface atomic-shells, while abrupt changes of the binding energy are always expected after an atomic shell is completed, since the atomic coordination number changes. As will be explained by the cluster-size dependence of the ionization potential (IP) the intensity maxima in the mass-spectra obtained from photoionization near the threshold should occur for half-filled atomic shells (or subatomic shells) and not for full ones, as expected from the abrupt changes in the binding-energy. Note, atomic shell structures have already previously been observed for inert gas clusters (4). In the case of metal clusters, typically 301
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P STAMPFLIAND KH BENNEMANN
a transition of the intensity oscillations in the mass-spectra due to electronic shells to a different structure due to atomic shells has been obtained (1). Finally, also supershell structure resulting from periodic bunching of energy levels has been reported (5). In the following we present a simple and unified physical picture for this seemingly general behavior of clusters. Furthermore, we present new results on the shell structure in the ionization-potential and photoyield and present an analytical theory for the shell slructure including the supershell structure. Our results should be useful for distinguishing between electronic- and atomic shell structure.
THEORY It is commonly believed that the features in the mass spectra (the sequence of magic numbers) reflect the stability of clusters, which is determined by the single electron energy spectrum el(n) and by the binding energy Eb(n) of the clusters (6). For small clusters mass spectra features are well characterized by the structure resulting from the specific electronic shells in the energy spectrum el(n) or corresponding structure in the electronic density of statesN(e) of the giant atom model. Of course, this sheO structure must disappear for larger clusters, since the electronic energy spectrum and N(e) becomes more and more continuous (An - n1/3). On the other hand, however, as long as the bulk coordination number or planar surface structure is not reached uniformly, the cohesive energy will continue to increase and there will be abrupt changes as one atomic shell or particular surface of the cluster is completed and one begins to add a new atom to the cluster (s. bond-energy picture and abrupt changes in the coordination number z) (7). Clearly, the transition from dominantly electronic shell to atomic shell determined magic numbers should happen at a certain cluster size (ncr), where Anbecomes smaller than the abrupt change in binding energy as one begins to fill a new atomic shell. Note, ncr is much smaller for crystalline-like than for liquid-like clusters for which the atomic surface structure is washed out. The situation for the size-dependence of the shell structure AEb(n) in the binding energy resulting from electronic- or atomic shells is illustrated in Figure 1. We conclude on general physical grounds that below the melting temperature, Tin, magic numbers should occur also for larger clusters due to atomic shell structure. Moreover, such a transition is also reflected in the ionization potential IPn, however, at cluster sizes ncr different from the ones referring to Eb(n), s. Figure 2, and furthermore also in the photoyield (Figure 3), cluster fragmentation, and other cluster properties. Thus, the structure in the mass spectra of clusters is best characterized by electronic shells in smaller clusters and by atomic shells in larger clusters (Figure 4). For a more quantitative description of our unified model for shell oscillations we proceed as follows. All shell structure (magic numbers) results in principle from the structure in the electronic density of states N(c) and are determined by the same expression
eb(n) = ~ d e ( e - e~)Nj(e) + AE(n)
[1]
J
for the binding energy (6,8). Here, Nile) denotes the electronic density of states of a cluster atom j with atomic states centered at ej. AE(n) results from remaining Coulomb interactions and from other repulsive interactions (Born-Meyer-type). AEb(n) can be determined rather accurately by using tight-binding type calculations (6,9). For simple metals with delocalized electrons and
UNIFIED MODEL FORTHE SHELLSTRUCTUREOF METALLICCLUSTERS
303
relatively weak interactions one may approximately determine Eb(n) for (smaller) clusters from using the giant atom model (2). The resultant shell (and supershell) structure in the energy spectrum or in Ni(e) leads then to relatively abrupt changes in Eb(n), when one starts to f'dl a new electronic shell (10). One has approximately for the abrupt energy change causing magic numbers Eb(n + 1) E~n) = A,, where An = {~:i+l(n+ 1) - E:i(n)} is the energy gap at eF, where el(n) is the energy of the state characterized by i. One obtains approximately (10) An - n q/3. For larger clusters the energy spectrum ¢i becomes nearly continuous (11). Then the abrupt energy changes aEb(n) still resulting from atomic structure must be determined directly from using Eq. [1]. -
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Figure 1. Shell structure energy AEt,(n). (a) refers to electronic shell structure and (b) to atomic shell structure. (c) Crossing of the structure energy due to electronic shells and atomic shells yielding ncr (- 40). Note, ncr depends somewhat on parameters, but in general ncr < 150.
304
P STAMPFLIAND KH BENNEMANN
RESULTS AND DISCUSSION First, we determine the structure due to electronic shells. This can be done rather accurately by extending (10) the theory for the supershell structure by Balm, Bloch and by Mottelson, Nishioka (5). The electronic density of states is given by N(e) = -N(e) + A(e), where N(e) refers to the average electronic density of states and the oscillating contribution is given by (5) AN(e) -• ,iAi(e, n)cos(kLi + ~). Here, the coefficients Ai are slowly varying functions of e and n and (kLi + ¢Pi)is the phase of a closed electron orbital of length Li = tx/R, R being the radius of the cluster. Assuming for spherical clusters that triangular and square orbitals contribute dominantly (5), we take for simplicity A3 = A4 = A/2 arid obtain then straightforwardly AN(e) -_-A(e, n)cos(kl +9)cos(kL +(~) with ! = (L4 - L3)/2 = 0.23rsn 113,L = (L4 + L3)12 = 5.4rsn 1/3, and 9 =(94 - ~ ) / 2 and ~ = (94 + ~ )/2. r s is the electronic radius. Here, the rapid oscillations due to cos( kL + ¢) define
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Figure 2. Results for the shell structure in the ionization potential IPn. (a) refers to structure due to electronic shells, and (b) refers to structure due to atomic shells. In (c) the crossing of the two AJP envelope curves is illustrated (~cr ~ 1000).
UNIFIED MODELFORTHE SHELLSTRUCTUREOF MEI'AI.UC CLUSTERS
305
the shells and the slower oscillations due to cos(kl + fp) the supershells generated by the interference of the i -- 3 and i-- 4 orbital. To determine now A we assume a potential well yielding N(e) = 2mRZ~t2f(2mR2itr2e).Here,f is an universal function (10). Comparing this with the above expression for AN(e), where A has been expanded in powers of e and n, we find Me, n ) ~ n 2(~ 1)/3e~, (R - n 113);and estimate ?--- 1/4. This allows then to determine in an analytical way the shell structure in contrast to previous work (5). From n = ~_e~(,,) deN(e) = ~e.£(t,) de-N(e), and thus eF(n) - tF(b) = - ('N (eF)) l j e_F(t,) deaN(e) - N(e)), we obtain the oscillations in the Fermi-energy, (after substituting the expression for AN(e) and integration), eF(n)----eF(b)-2-0.26CeF(b)n2¢'t|)13sin(kFL + ¢)cos(kFl + qO.
[2]
Similarly, for the shell structure in the binding energy E~,l = f e£ ( N ( e ) --N(e))ede one gets the physically expected result (0 / On)AE~l(n) ~ -(eF(n)- eF(b)),
[3]
AE~,t ( n ) = O. 074 CeF( b )n27/3cos( kFL + (b)cos( kFl + ~p).
[4]
and then
This gives E~t(n) ~ n 116, forT= 1/4. Results for E~,/obtained by using Eq. [4] are shown in Figure 1(a). Our results compare well with those by Nishioka (5). if we use ¢ = 2.91 (implying (9 ~-0.4), and C = 2. This is very remarkable in view of the heavy numerical work necessary for the previous analysis (5). Note, ¢ and q) could be calculated from the self-consistent potential of the giant atom model. Secondly, we determine the oscillations in the binding energy resulting from atomic cluster shells. Note the effect of the atomic shells on the cohesive energy results from the varying coordination numbers zi of the individual atoms i. According to tight-binding and effective medium theories the binding energy of simple metals is roughly proportional to the square root of the coordination number and thus we estimate the shell structure
l
where Zb and E6 are the bulk coordination number and energy, respectively, and S the surface energy per unit area. The results shown in Figure l(b) for Nan clusters are obtained assuming a structure of concentric icosahedral layers and a random atomic occupation of the topmost layer. Depending on the surface structure, AEb(n) may vary gradually in the open shell, but also somewhat abruptly yielding smaller subshell structure. This may be the case, if the cluster has different surface faces or if the outer atomic shell of the cluster is not randomly filled. The jumps AEb(n) are different for more open cluster surfaces (s. b.c.c.-like cluster structures vs. f.c.c, ones, etc.), or for Tbelo w or above Tm( n ). S ince AEo( n ) depends on the surface structure, it should reflec t also structural transitions caused by temperature or increasing cluster size. The magnitude of the shell effect is proportional to the surface of the cluster (corresponding to the number of atoms in
306
P STAMPFLIAND KH BENNEMANN
0100 lVv .-
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I
,
0
,
1000
,
2000
,
3000
4000
5000 n
Figure 3. Results for the photoyield given by Yn ~ (ho~ - IPn) 2. (a) refers to electronic shell structure; (b) refers to atomic shell structure. Note, for atomic shells we use a somewhat lower photon energy ~ than for electronic shells. For ALP,, we use the results shown in Figure 2. one shell). This leads to a more rapid increase of the shell effect for atomic shells (E~,t(n) - n 2/3) in comparison to electronic shells. Note, that qualitatively similar results are expected for more accurate models of the binding energy and growth of atomic layers. Comparing the shell structure in the binding energy AEt,(n) due to electronic and atomic shells we estimate from AE~,l = AE~t the transition from electronic shell to atomic shell structure, s. Figure 1(c). This transition occurs at a critical cluster size ncr which depends on cluster structure and temperature. To observe most prominently electronic (super) shell structure one needs liquid800
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b-, 400 o~o~
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~
200 atomic shell structure
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40 nl/3
Figure 4. Illustration of electronic- vs. atomic-shell structure in the (T, n) plane. Approximately, below the melting temperature Tm atomic shell structure dominates due to dephasing of electronic orbits. For high T the electronic shell structure is washed out and thus vanishes.
UNIFIED MODEL FORTHE SHELL STRUCTUREOF METALLICCLUSTERS
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Figure 5. Temperature dependence of the electronic shell structure in the Gibbs free energy AG(T,n). AG(T,n) ~ AE~I(n) for T --~ 0. Note that 400 K is just above the melting .temperature of sodium. The dotted lines indicate the amplitude of the shell effect at T = 0 (see Figure 4a). like cluster structures (T> T,,,, T,,, = melting temperature) which is difficult to achieve for larger clusters with a large bulk T,,(b) (approximately Tin(n) - ~ / z b T,,,(b)). If the cluster has solid structure, its shape becomes a polyhedron (icosahedron or cubooctahedron). Then. if the dephasing due to the different lengths L of the distinct triangular (or square) orbitals becomes too large, AL kF > 2n, the electronic shell structure disappears. This is illustrated in Figure 4. To understand important recent experimental results (1), we note that the shell structure oscillations in [eF(n)--eF(b)], Eq. [2], cause also oscillations in the ionization potential IPn, since IPn =-IPb + e2/(2R ) + ALP,,, with AlPn =-[eF(n) - eF(b)] for the electronic shell structure and ALP,, = -4cx(1-x ) for atomic shell structure. Here, x = (n - n i ) / ( n i + ! - ni) measures the atomic filling of the (i + 1)-th cluster shell with ( n i + l - n i ) atomic sites. This expression forAlPn yields a minimum o f l P for half-filled atomic shell and is suggested by the fact that rough surfaces (corresponding to incomplete atomic cluster shells) have a smaUer IP than compact surfaces (12). We estimate c = 0.1eV. Results for IP, are shown in Figures 2. Comparing Figures 1 and 2 one notes that the oscillations in E~t and IPn are shifted relative to each other by (k/4), where K refers to the "wavelength" of the rapid oscillations. Minima in IPn occur approximately at the beginning of an electronic shell filling and occur for half-t'dled atomic shells with maximal surface roughening. Now, the shell structure effects in IPn determine the intensity of the mass spectrum observed by Martin et al. (1) for larger clusters ionized by photoionization near threshold. Intensity maxima are observed for half-filled atomic shells in contrast to what is expected from the oscillation in the
308
P STAMPFLIANDKH BENNEMANN
I
0.3
I
I
T=400 K
0.2 J
P..
I
0.1 0
-o
! 0
I
I
I
I
5
10
15
20
25 n ~/3
Figure 6. Temperature dependence of the electronic shell structure in the ionization potential AIP(T, n). The dotted lines indicate the amplitude of the shell effect at T = 0 (see Figure 2a). binding e n e r g y AUgt . TOexplain this shift we note that the intensity of the mass spectrum observed by photoionization near threshold is determined by the shell structure in IPn. This can be estimated from In+(hog) = crn(hco)I °, where In° and In+ refer to the intensity of neulral and ionized clusters and [t0~ is the photon energy. We use the Fowler-Nordheim law for the photoyield Yn. Thus, (~,o >-IPn) Y n - ([l(0 - I e n )2 ~ O.n (h0.)) / g 2 ~
In+ [ In° ,
[6]
where an denotes the ionization cross section. It follows immediately from Eq. [6] that a minimum in IPn yields a maximum for Yn and the intensity In+ and which does not correspond to a maximum of I ° or Eb(n). Note, oscillations in In correspond of course to those in Eb(n). For much larger photon energies ho) one has on(hw) = const, and then one has a correspondence of the oscillations in Yn and I ° or Eb(n). In Figure 3 results are given for Yn which are in close agreement with the experimental observations (1). This completes then the explanation for the observed shift in the shell structure oscillations in In+ and Eb(n). Note, asymptotically zSdPn - n -112 in the case of electronic shells and AIPn ~ const in the case of atomic shells. The transition from oscillations dominantly due to electronic shells to ones due to atomic shells occurs for different cluster sizes (ncr- 1000) than is the case for Eb(n). In Figure 4 we estimate the temperature dependence of the shell structure. From n > l'lcron the atomic shell structure dominates below the (estimated) melting temperature Tin. Above Tm the electronic shell structure is expected to dominate, if the atomic structure is washed out sufficiently.
UNtFIED MODELFORTHE SHELLSTRUCTUREOF METALLICCLUSTERS
309
Then, for T > Tm the electronic shell structure will disappear at ncr(T) for increasing cluster sizes, since the energy spectrum of the giant atom modelling the cluster gets continuous and smeared out. This is calculated from ( ~ n ) - ~ b ) ) (10). Results for the temperature dependence of the electronic shell effect are shown in Figures 5 and 6. Note, the effect of f'mite temperatures Tbecomes stronger for increasing cluster size (teff 0, T nil3), thus the shell structure in the cohesive energy decreases for very large cluster sizes n instead of a further increase *~ n]/6 as expected for T= 0. In addition, magnetic fields should have a strong effect on the electronic shell structure. If the cyclotron radius of the electron is of similar (or smaller) magnitude as the diameter of the cluster, then the triangular and square paths are effectively destroyed and the electronic shell structure is strongly attenuated, in contrast to the atomic shell structure, which is not affected. CONCLUSION In summary, we have presented an analytical theory for shell effects in the binding energy, ionization-potential, and photoyield. The results presented should be useful for distinguishing atomic shell structure effects from electronic shell structure ones. REFERENCES 1. 2.
3. 4. 5.
6.
7. 8. 9. 10.
11.
12.
T.P. Martin, T. Bergmann, H. GOhlich, and T. Lange, Z. Phys. D19, 25 (1991). W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 52, 2141 (1984); J.L. Martin, R. Car, J. Buttet, Surf. Sci. 106, 265 (1981); W. Ekardt, Ber. Bunsenges. Phys. Chem. 88, 289 (1984); Phys. Rev. B 29,1558 (1984). T.P. Martin, T. Bergmarm, H. Gohlich, T. Lange, Chem. Phys. Lett. 172,209 (1990); S. Bjornholm, J. Borggren, O. Echt, K. Hansen, J. Pedersen, H.D. Rasmussen, Phys. Rev. Lett. 65,1627 (1990). O. Echt, K. Sattler, E. Recknagel, Phys. Rev. Lett. 47, 1121 (1981); W. Miehle, D. Kandler, T. Leisner, O. Echt, J. Chem. Phys. 91, 5940 (1989). M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (Springer New York, 1990); H. Nishioka, Z. Phys. D 19, 19 (1991); H. Nishioka, K. Hansen, and B.R. Mottelson, Phys. Rev. B 42, 9377 (1990); R. Balian, C. Bloch, Ann. Physik 69, 76 (1971). D. TomAnek, S. Mukherjee, and K.H. Bennemann, Phys. Rev. B 28, 665 (1983). The quantum number i characterizing El(n) refers to the main quantum number and the angular momentum in the case of the giant atom model. Note, these abrupt changes AEb(n) may be related to chemisorption energies (for example Na on Nan, etc.) and will depend on the atomic structure of the clusters and its various surface faces. D. TomL'aek, V. Kumar, S. Holloway, and K.H. Bennemann, Sol. St. Commun. 41, 27 (1982). M. Manninen, J. Mansikka-Aho, and E. Hammar'en, Europhys. Lett. 15,423 (1991). P. Stampfli, andK.H. Bennemann, ZfurPhysikD25,87(1992) andPhys. Rev. Lett. 69,3471 (1992). At f'mite temperature one has Ala(n) - [N(eF )V l f de.f(e)(N(e)-N(e)) where ~ is the chemical potential andfis the Fermi function, and AEb should he replaced by the free energy AFt,, s. O. Genzken and M. Brack, Phys. Rev. Lett. 67,3286 (1991). Note, one estimates for Na-like clusters around the highest occupied electronic levels (eF) a level spacing of about 0.SeV for clusters having about n ~ 100 atoms (and approximately a level spacing of about 0.05eV for larger cluster having 1000 or more atoms). This would yield ncr of the order of 100 atoms. R. Smoluchowski, Phys. Rev. 60, 661 (1941). Note, for Na IP _=2.65eV for (111)-, IP = 2.75eV for (100)-, and IP _=_3.10eV for (110)-surface.
0965-9773/93 $6.00 + .00 PRINTEDIN THE USA
UNIFIED MODEL FOR THE ELECTRONIC AND ATOMIC SHELL STRUCTURES OF METALLIC CLUSTERS
P. Stampfli and K.H. Bennemann Institute for Theoretical Physics, Freie Universit~t Berlin Arnimallee 14, W-1000 Berlin 33, Germany
Abstract--We present a theory explaining from a general point of view the intensity anomalies (magic numbers) in the mass spectra of metallic clusters. The transition from an electronic shell dominated structure in the mass spectra of smaller clusters to an atomic shell dominated structure in the mass spectra of larger clusters is discussed as a function of cluster size n and temperature T. We estimate where in the (T, n)plane electronic- and where atomic shell structure dominates. Results are given for the shell structure in the cohesive energy, the ioni:ation potential and photoyield, which are useful for distinguishing between the two kinds of structure. INTRODUCTION The mass spectra of metallic clusters have been for many years an interesting tool for learning about the electronic and atomic structure of metallic clusters (1). The mass-intensity spectrum observed for smaller clusters of many metal-atoms led to the invention of the giant atom model for clusters and the resultant electronic spectrum with characteristic electronic shell structure (2). The electronic energy spectrum was characterized by shells. These consist of several energy levels el, which are nearly degenerate although they have different angular momenta 1. Peaks in the mass spectrum resulted when the difference An = (En -En-1) between cluster energies En changed abruptly, as is the case when a shell is f'dled by electrons and the next higher energy levels are separated by a gap. Thus, the essential features of the mass spectra of many smaller metallic clusters could be explained (3). Recently, intensity anomalies (magic numbers) of larger clusters (s. Nan: 1500 < n < 22000) have been observed and interpreted as resulting from an atomic shell structure of the metallic clusters (1), Intensity maxima of ionized clusters resulting from photoionization have been observed at half-filled surface atomic-shells, while abrupt changes of the binding energy are always expected after an atomic shell is completed, since the atomic coordination number changes. As will be explained by the cluster-size dependence of the ionization potential (IP) the intensity maxima in the mass-spectra obtained from photoionization near the threshold should occur for half-filled atomic shells (or subatomic shells) and not for full ones, as expected from the abrupt changes in the binding-energy. Note, atomic shell structures have already previously been observed for inert gas clusters (4). In the case of metal clusters, typically 301
302
P STAMPFLIAND KH BENNEMANN
a transition of the intensity oscillations in the mass-spectra due to electronic shells to a different structure due to atomic shells has been obtained (1). Finally, also supershell structure resulting from periodic bunching of energy levels has been reported (5). In the following we present a simple and unified physical picture for this seemingly general behavior of clusters. Furthermore, we present new results on the shell structure in the ionization-potential and photoyield and present an analytical theory for the shell slructure including the supershell structure. Our results should be useful for distinguishing between electronic- and atomic shell structure.
THEORY It is commonly believed that the features in the mass spectra (the sequence of magic numbers) reflect the stability of clusters, which is determined by the single electron energy spectrum el(n) and by the binding energy Eb(n) of the clusters (6). For small clusters mass spectra features are well characterized by the structure resulting from the specific electronic shells in the energy spectrum el(n) or corresponding structure in the electronic density of statesN(e) of the giant atom model. Of course, this sheO structure must disappear for larger clusters, since the electronic energy spectrum and N(e) becomes more and more continuous (An - n1/3). On the other hand, however, as long as the bulk coordination number or planar surface structure is not reached uniformly, the cohesive energy will continue to increase and there will be abrupt changes as one atomic shell or particular surface of the cluster is completed and one begins to add a new atom to the cluster (s. bond-energy picture and abrupt changes in the coordination number z) (7). Clearly, the transition from dominantly electronic shell to atomic shell determined magic numbers should happen at a certain cluster size (ncr), where Anbecomes smaller than the abrupt change in binding energy as one begins to fill a new atomic shell. Note, ncr is much smaller for crystalline-like than for liquid-like clusters for which the atomic surface structure is washed out. The situation for the size-dependence of the shell structure AEb(n) in the binding energy resulting from electronic- or atomic shells is illustrated in Figure 1. We conclude on general physical grounds that below the melting temperature, Tin, magic numbers should occur also for larger clusters due to atomic shell structure. Moreover, such a transition is also reflected in the ionization potential IPn, however, at cluster sizes ncr different from the ones referring to Eb(n), s. Figure 2, and furthermore also in the photoyield (Figure 3), cluster fragmentation, and other cluster properties. Thus, the structure in the mass spectra of clusters is best characterized by electronic shells in smaller clusters and by atomic shells in larger clusters (Figure 4). For a more quantitative description of our unified model for shell oscillations we proceed as follows. All shell structure (magic numbers) results in principle from the structure in the electronic density of states N(c) and are determined by the same expression
eb(n) = ~ d e ( e - e~)Nj(e) + AE(n)
[1]
J
for the binding energy (6,8). Here, Nile) denotes the electronic density of states of a cluster atom j with atomic states centered at ej. AE(n) results from remaining Coulomb interactions and from other repulsive interactions (Born-Meyer-type). AEb(n) can be determined rather accurately by using tight-binding type calculations (6,9). For simple metals with delocalized electrons and
UNIFIED MODEL FORTHE SHELLSTRUCTUREOF METALLICCLUSTERS
303
relatively weak interactions one may approximately determine Eb(n) for (smaller) clusters from using the giant atom model (2). The resultant shell (and supershell) structure in the energy spectrum or in Ni(e) leads then to relatively abrupt changes in Eb(n), when one starts to f'dl a new electronic shell (10). One has approximately for the abrupt energy change causing magic numbers Eb(n + 1) E~n) = A,, where An = {~:i+l(n+ 1) - E:i(n)} is the energy gap at eF, where el(n) is the energy of the state characterized by i. One obtains approximately (10) An - n q/3. For larger clusters the energy spectrum ¢i becomes nearly continuous (11). Then the abrupt energy changes aEb(n) still resulting from atomic structure must be determined directly from using Eq. [1]. -
I
I
.
1/6
a)
n,N
I
..............................
> v
-1
-2-
b)
3O
>
L
I n 2/3
I
,°°"
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N o -10 -20-30-
I ,-. :>
43-
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. I //atomic
I shell
113/
2 1
0
l
0
5
i
10
20
15 nl/3
Figure 1. Shell structure energy AEt,(n). (a) refers to electronic shell structure and (b) to atomic shell structure. (c) Crossing of the structure energy due to electronic shells and atomic shells yielding ncr (- 40). Note, ncr depends somewhat on parameters, but in general ncr < 150.
304
P STAMPFLIAND KH BENNEMANN
RESULTS AND DISCUSSION First, we determine the structure due to electronic shells. This can be done rather accurately by extending (10) the theory for the supershell structure by Balm, Bloch and by Mottelson, Nishioka (5). The electronic density of states is given by N(e) = -N(e) + A(e), where N(e) refers to the average electronic density of states and the oscillating contribution is given by (5) AN(e) -• ,iAi(e, n)cos(kLi + ~). Here, the coefficients Ai are slowly varying functions of e and n and (kLi + ¢Pi)is the phase of a closed electron orbital of length Li = tx/R, R being the radius of the cluster. Assuming for spherical clusters that triangular and square orbitals contribute dominantly (5), we take for simplicity A3 = A4 = A/2 arid obtain then straightforwardly AN(e) -_-A(e, n)cos(kl +9)cos(kL +(~) with ! = (L4 - L3)/2 = 0.23rsn 113,L = (L4 + L3)12 = 5.4rsn 1/3, and 9 =(94 - ~ ) / 2 and ~ = (94 + ~ )/2. r s is the electronic radius. Here, the rapid oscillations due to cos( kL + ¢) define
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> C o-
C
C) >
~ *
v
elec. shell t
\
'
1/3
0.1 ~"-atomic shell ~ 0 0
lb
1'5
20 n~/3
Figure 2. Results for the shell structure in the ionization potential IPn. (a) refers to structure due to electronic shells, and (b) refers to structure due to atomic shells. In (c) the crossing of the two AJP envelope curves is illustrated (~cr ~ 1000).
UNIFIED MODELFORTHE SHELLSTRUCTUREOF MEI'AI.UC CLUSTERS
305
the shells and the slower oscillations due to cos(kl + fp) the supershells generated by the interference of the i -- 3 and i-- 4 orbital. To determine now A we assume a potential well yielding N(e) = 2mRZ~t2f(2mR2itr2e).Here,f is an universal function (10). Comparing this with the above expression for AN(e), where A has been expanded in powers of e and n, we find Me, n ) ~ n 2(~ 1)/3e~, (R - n 113);and estimate ?--- 1/4. This allows then to determine in an analytical way the shell structure in contrast to previous work (5). From n = ~_e~(,,) deN(e) = ~e.£(t,) de-N(e), and thus eF(n) - tF(b) = - ('N (eF)) l j e_F(t,) deaN(e) - N(e)), we obtain the oscillations in the Fermi-energy, (after substituting the expression for AN(e) and integration), eF(n)----eF(b)-2-0.26CeF(b)n2¢'t|)13sin(kFL + ¢)cos(kFl + qO.
[2]
Similarly, for the shell structure in the binding energy E~,l = f e£ ( N ( e ) --N(e))ede one gets the physically expected result (0 / On)AE~l(n) ~ -(eF(n)- eF(b)),
[3]
AE~,t ( n ) = O. 074 CeF( b )n27/3cos( kFL + (b)cos( kFl + ~p).
[4]
and then
This gives E~t(n) ~ n 116, forT= 1/4. Results for E~,/obtained by using Eq. [4] are shown in Figure 1(a). Our results compare well with those by Nishioka (5). if we use ¢ = 2.91 (implying (9 ~-0.4), and C = 2. This is very remarkable in view of the heavy numerical work necessary for the previous analysis (5). Note, ¢ and q) could be calculated from the self-consistent potential of the giant atom model. Secondly, we determine the oscillations in the binding energy resulting from atomic cluster shells. Note the effect of the atomic shells on the cohesive energy results from the varying coordination numbers zi of the individual atoms i. According to tight-binding and effective medium theories the binding energy of simple metals is roughly proportional to the square root of the coordination number and thus we estimate the shell structure
l
where Zb and E6 are the bulk coordination number and energy, respectively, and S the surface energy per unit area. The results shown in Figure l(b) for Nan clusters are obtained assuming a structure of concentric icosahedral layers and a random atomic occupation of the topmost layer. Depending on the surface structure, AEb(n) may vary gradually in the open shell, but also somewhat abruptly yielding smaller subshell structure. This may be the case, if the cluster has different surface faces or if the outer atomic shell of the cluster is not randomly filled. The jumps AEb(n) are different for more open cluster surfaces (s. b.c.c.-like cluster structures vs. f.c.c, ones, etc.), or for Tbelo w or above Tm( n ). S ince AEo( n ) depends on the surface structure, it should reflec t also structural transitions caused by temperature or increasing cluster size. The magnitude of the shell effect is proportional to the surface of the cluster (corresponding to the number of atoms in
306
P STAMPFLIAND KH BENNEMANN
0100 lVv .-
lsh
o.08-
0.060.040.020.00
I
,
0
,
1000
,
2000
,
3000
4000
5000 n
Figure 3. Results for the photoyield given by Yn ~ (ho~ - IPn) 2. (a) refers to electronic shell structure; (b) refers to atomic shell structure. Note, for atomic shells we use a somewhat lower photon energy ~ than for electronic shells. For ALP,, we use the results shown in Figure 2. one shell). This leads to a more rapid increase of the shell effect for atomic shells (E~,t(n) - n 2/3) in comparison to electronic shells. Note, that qualitatively similar results are expected for more accurate models of the binding energy and growth of atomic layers. Comparing the shell structure in the binding energy AEt,(n) due to electronic and atomic shells we estimate from AE~,l = AE~t the transition from electronic shell to atomic shell structure, s. Figure 1(c). This transition occurs at a critical cluster size ncr which depends on cluster structure and temperature. To observe most prominently electronic (super) shell structure one needs liquid800
I
I
I
I
~ n o electronic shell ~ structure
,.. 600
I
|
I
structure
b-, 400 o~o~
I T m
~
200 atomic shell structure
0
lb
40 nl/3
Figure 4. Illustration of electronic- vs. atomic-shell structure in the (T, n) plane. Approximately, below the melting temperature Tm atomic shell structure dominates due to dephasing of electronic orbits. For high T the electronic shell structure is washed out and thus vanishes.
UNIFIED MODEL FORTHE SHELL STRUCTUREOF METALLICCLUSTERS
3
;>
/
I
I
t
307
I
2 t T=400 K "
.
, * • .- .. •
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' """ """" ""
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•""
-2
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.o. OS "'-..-"
-3 0
, 5
, 10
, 15
, 20
25 n 1/3
Figure 5. Temperature dependence of the electronic shell structure in the Gibbs free energy AG(T,n). AG(T,n) ~ AE~I(n) for T --~ 0. Note that 400 K is just above the melting .temperature of sodium. The dotted lines indicate the amplitude of the shell effect at T = 0 (see Figure 4a). like cluster structures (T> T,,,, T,,, = melting temperature) which is difficult to achieve for larger clusters with a large bulk T,,(b) (approximately Tin(n) - ~ / z b T,,,(b)). If the cluster has solid structure, its shape becomes a polyhedron (icosahedron or cubooctahedron). Then. if the dephasing due to the different lengths L of the distinct triangular (or square) orbitals becomes too large, AL kF > 2n, the electronic shell structure disappears. This is illustrated in Figure 4. To understand important recent experimental results (1), we note that the shell structure oscillations in [eF(n)--eF(b)], Eq. [2], cause also oscillations in the ionization potential IPn, since IPn =-IPb + e2/(2R ) + ALP,,, with AlPn =-[eF(n) - eF(b)] for the electronic shell structure and ALP,, = -4cx(1-x ) for atomic shell structure. Here, x = (n - n i ) / ( n i + ! - ni) measures the atomic filling of the (i + 1)-th cluster shell with ( n i + l - n i ) atomic sites. This expression forAlPn yields a minimum o f l P for half-filled atomic shell and is suggested by the fact that rough surfaces (corresponding to incomplete atomic cluster shells) have a smaUer IP than compact surfaces (12). We estimate c = 0.1eV. Results for IP, are shown in Figures 2. Comparing Figures 1 and 2 one notes that the oscillations in E~t and IPn are shifted relative to each other by (k/4), where K refers to the "wavelength" of the rapid oscillations. Minima in IPn occur approximately at the beginning of an electronic shell filling and occur for half-t'dled atomic shells with maximal surface roughening. Now, the shell structure effects in IPn determine the intensity of the mass spectrum observed by Martin et al. (1) for larger clusters ionized by photoionization near threshold. Intensity maxima are observed for half-filled atomic shells in contrast to what is expected from the oscillation in the
308
P STAMPFLIANDKH BENNEMANN
I
0.3
I
I
T=400 K
0.2 J
P..
I
0.1 0
-o
! 0
I
I
I
I
5
10
15
20
25 n ~/3
Figure 6. Temperature dependence of the electronic shell structure in the ionization potential AIP(T, n). The dotted lines indicate the amplitude of the shell effect at T = 0 (see Figure 2a). binding e n e r g y AUgt . TOexplain this shift we note that the intensity of the mass spectrum observed by photoionization near threshold is determined by the shell structure in IPn. This can be estimated from In+(hog) = crn(hco)I °, where In° and In+ refer to the intensity of neulral and ionized clusters and [t0~ is the photon energy. We use the Fowler-Nordheim law for the photoyield Yn. Thus, (~,o >-IPn) Y n - ([l(0 - I e n )2 ~ O.n (h0.)) / g 2 ~
In+ [ In° ,
[6]
where an denotes the ionization cross section. It follows immediately from Eq. [6] that a minimum in IPn yields a maximum for Yn and the intensity In+ and which does not correspond to a maximum of I ° or Eb(n). Note, oscillations in In correspond of course to those in Eb(n). For much larger photon energies ho) one has on(hw) = const, and then one has a correspondence of the oscillations in Yn and I ° or Eb(n). In Figure 3 results are given for Yn which are in close agreement with the experimental observations (1). This completes then the explanation for the observed shift in the shell structure oscillations in In+ and Eb(n). Note, asymptotically zSdPn - n -112 in the case of electronic shells and AIPn ~ const in the case of atomic shells. The transition from oscillations dominantly due to electronic shells to ones due to atomic shells occurs for different cluster sizes (ncr- 1000) than is the case for Eb(n). In Figure 4 we estimate the temperature dependence of the shell structure. From n > l'lcron the atomic shell structure dominates below the (estimated) melting temperature Tin. Above Tm the electronic shell structure is expected to dominate, if the atomic structure is washed out sufficiently.
UNtFIED MODELFORTHE SHELLSTRUCTUREOF METALLICCLUSTERS
309
Then, for T > Tm the electronic shell structure will disappear at ncr(T) for increasing cluster sizes, since the energy spectrum of the giant atom modelling the cluster gets continuous and smeared out. This is calculated from ( ~ n ) - ~ b ) ) (10). Results for the temperature dependence of the electronic shell effect are shown in Figures 5 and 6. Note, the effect of f'mite temperatures Tbecomes stronger for increasing cluster size (teff 0, T nil3), thus the shell structure in the cohesive energy decreases for very large cluster sizes n instead of a further increase *~ n]/6 as expected for T= 0. In addition, magnetic fields should have a strong effect on the electronic shell structure. If the cyclotron radius of the electron is of similar (or smaller) magnitude as the diameter of the cluster, then the triangular and square paths are effectively destroyed and the electronic shell structure is strongly attenuated, in contrast to the atomic shell structure, which is not affected. CONCLUSION In summary, we have presented an analytical theory for shell effects in the binding energy, ionization-potential, and photoyield. The results presented should be useful for distinguishing atomic shell structure effects from electronic shell structure ones. REFERENCES 1. 2.
3. 4. 5.
6.
7. 8. 9. 10.
11.
12.
T.P. Martin, T. Bergmann, H. GOhlich, and T. Lange, Z. Phys. D19, 25 (1991). W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 52, 2141 (1984); J.L. Martin, R. Car, J. Buttet, Surf. Sci. 106, 265 (1981); W. Ekardt, Ber. Bunsenges. Phys. Chem. 88, 289 (1984); Phys. Rev. B 29,1558 (1984). T.P. Martin, T. Bergmarm, H. Gohlich, T. Lange, Chem. Phys. Lett. 172,209 (1990); S. Bjornholm, J. Borggren, O. Echt, K. Hansen, J. Pedersen, H.D. Rasmussen, Phys. Rev. Lett. 65,1627 (1990). O. Echt, K. Sattler, E. Recknagel, Phys. Rev. Lett. 47, 1121 (1981); W. Miehle, D. Kandler, T. Leisner, O. Echt, J. Chem. Phys. 91, 5940 (1989). M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (Springer New York, 1990); H. Nishioka, Z. Phys. D 19, 19 (1991); H. Nishioka, K. Hansen, and B.R. Mottelson, Phys. Rev. B 42, 9377 (1990); R. Balian, C. Bloch, Ann. Physik 69, 76 (1971). D. TomAnek, S. Mukherjee, and K.H. Bennemann, Phys. Rev. B 28, 665 (1983). The quantum number i characterizing El(n) refers to the main quantum number and the angular momentum in the case of the giant atom model. Note, these abrupt changes AEb(n) may be related to chemisorption energies (for example Na on Nan, etc.) and will depend on the atomic structure of the clusters and its various surface faces. D. TomL'aek, V. Kumar, S. Holloway, and K.H. Bennemann, Sol. St. Commun. 41, 27 (1982). M. Manninen, J. Mansikka-Aho, and E. Hammar'en, Europhys. Lett. 15,423 (1991). P. Stampfli, andK.H. Bennemann, ZfurPhysikD25,87(1992) andPhys. Rev. Lett. 69,3471 (1992). At f'mite temperature one has Ala(n) - [N(eF )V l f de.f(e)(N(e)-N(e)) where ~ is the chemical potential andfis the Fermi function, and AEb should he replaced by the free energy AFt,, s. O. Genzken and M. Brack, Phys. Rev. Lett. 67,3286 (1991). Note, one estimates for Na-like clusters around the highest occupied electronic levels (eF) a level spacing of about 0.SeV for clusters having about n ~ 100 atoms (and approximately a level spacing of about 0.05eV for larger cluster having 1000 or more atoms). This would yield ncr of the order of 100 atoms. R. Smoluchowski, Phys. Rev. 60, 661 (1941). Note, for Na IP _=2.65eV for (111)-, IP = 2.75eV for (100)-, and IP _=_3.10eV for (110)-surface.