- Email: [email protected]
Collective resonance frequencies in metal clusters
Volume 133, number 1,2
PHYSICS LETTERS A
31 October 1988
COLLECI1VE RESONANCE FREQUENCIES IN METAL CLUSTERS Vitaly KR~SIN Department ~f Physics,
University of California, Berkeley, CA 94720, USA
Received 7 J~.i1y1988; revised manuscript received 24 August 1988; accepted for publication 25 August 1988 Communicaledby A.A. Maradudin
The collective resonance frequency of valence electrons in small metal clusters is calculated analytically. The self-consistent solution takds into account the density profile of the valence electrons, as given by the recently developed statistical approach. The resonance frequency is red shifted with respect to its classical value, and depends on the amount ofthe electron cloud’s spillout outside the boundary of the positive cluster core.
The visible spectrum of small metal particles and clusters is domjnated by the surface plasma resonance of the val~nceelectrons (see, e.g., refs. [1—3]). For metal spheres of radius much less than the wavelength of incide~itlight, the classical theory ,[4] predicts a resonance at the frequency w~= ~ where air, is the bulk plasma frequency. Various calculations have predicted both a red shift and a blue shift of the resonance compared to the classical value, depending on the ~nodelused (see e.g. the review [1], and the recent flumerical calculations [51). In recent experiment~on small sodium clusters [2], a red shift was obser%~ed. Here we present results of an analytical calculation of the surface i~esonancefrequencies, applicable to all small metal diusters in which optical properties in the visible rang~are dominated by delocalized valence electrons. We solve the integral equation for the resonance fleld [61 and utilize the recently developed Thoma~—Fermi(TF) statistical description of metal clusterS [7]. The position of the resonance is found to be 4etermined by the density profile of the valence ele~trons.The spill-out of the electron cloud outside flue boundary of the positive core of the cluster lead~to a red shift of the resonance; the magnitude of t~ieshift depends sensitively on the amount of the spill-out. The latter is calculated using the aforementioned statistical method, which describes the electron density distribution in clusters, This method has been successful in treating cluster
properties governed by the valence electrons, such as relative stabilities, ionization potentials and diamagnetism [7]. The general integral equation describing self-supporting oscillations in an electron gas in the randomphase approximation is (see, e.g., ref. [81) V(r) =e2
J
d3r 1 r— r1 I —
X
~
w~(r~ )~ti~(r1) ~ A A
A
V~ ,
(1)
A —
where V(r) is the field inside the system; ~A and P~VA are the single-particle energies and wave functions, respectively. The frequency w. for which a non-trivial solution exists is the resonance frequency. In ref. [61 it is shown that this equation can be brought into the following form: (1_ 4En(r) e~ V(r) \. mw~ ) 2 3r = — ci 1 V(r1 ) v1 1 mw~ Ir—r1 _-~_-~-
J
v1 n(r1) (2)
where n (r) is the number density of valence electrons in the particle. The authors substitute the stepfunction profile n (r) = nO(R — r) into eq. (2) and recover the classical surface plasmon frequency, w~.
0375-9601 1881$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
89
Volume 133, number 1,2
PHYSICS LETTERS A
For small clusters, it is essential to take into account the correct form of the electron density distribution n (r). For metal clusters, the TF statistical method [7] can be used to describe this quantity. Consider the case of spherical metal clusters (i.e. the “magic number” clusters with spherical closed shells [9]. The positive ion background of the cluster is treated as a uniform sphere of radius R = 3(a aor~N~/ 0 is the Bohr radius, r~is the bulk Wigner—Seitz density parameter and Na is the number of atoms in the cluster), The function n ( r) decreases rapidly in the region r~R [7], and its derivative is therefore sharply peaked in this region. This fact allows eq. (2) to be solved for ~ analytically. After some calculations, we find that the ratio of the resonance frequency to the classical surface plasmon frequency is given by 23g(R)+l
2(a)r/ws)
—{[3g(R)— 1 ]2+24q(l
—
3q)} 1/2,
(3)
where q=~Jg(r)dr,
k6
(r/R)
3/2
—
(r/R
—
D)
6,
(5)
where k=(l2/it)3r~”2N,Y3 D=l—
61 k
12 — —~--
k
+
...
,
(6)
and Ne is the number of valence electrons in the cluster. For a chosen cluster size and material, it is thus possible to calculate the resonance frequency. For instance, for Na 8, we find that the ratio ofthe latter to the classical surface plasmon frequency is WrI = 0.74; for Na20 it is 0.79, and for Na92, 0.84. (Recent experiments [2] have shown that for Na8 this
90
ratio is 0.75 ±0.02; our theory is in excellent agreement with these data.) To summarize, we have calculated the collective resonance frequencies of small metal clusters. This was done by solving the RPA integral equation [61 and accurately taking into account the density distribution of valence electrons in clusters. The results ofthe TF statistical method [7] are used for the density. The spill-out of the electron cloud inside the boundary of the positive background is shown to lead to a red shift in the position of the resonance, and the magnitude of the shift is directly related to the amount of the spill-out. This approach can be extended to calculate the static electric polarizability of metal clusters; this calculation will be presented elsewhere. I would like to thank Professor W.D. Knight, Dr. V.Z. Kresin and Kathy Selby for very useful discussions. This work was supported by the U.S. National Science Foundation under Grant No. DMR-8615246.
(4)
and g(r) is the dimensionless electron density profile defined by n(r)=p~g(r),p~being the number density of the uniform positive background charge. According to the TF solution [7], the function g( r) for r~R is given by g( r) =
31 October 1988
References [1] U. Kreibig and L. Genzel, Surf. Sci. 156 (1985) 678. [2] W.A. de Heer, K. Selby, V. Kresin, J. Masui, M. Vollmer, A. Châtelain and W.D. Knight, Phys. Rev. Lett. 59 (1987)1805; K. Selby, M. Voilmer, J. Masui, V. Kresin, W.A. de Heer and W.D. Knight, in: 4th mt. Symp. on Small particles and inorganic clusters, to be published in Z. Phys., K. Selby, M. Vollmer, J. Masui, V. Kresin, W.A. de Heer and W.D. Knight, to be published. [3] W. Hoheisel, K. Jungmann, M. Voilmer, R. Weidenauer and F.Trager,Phys.Rev.Lett.60(l988) 1649. [4] G.Mie,Ann.Phys.25 (1908) 377. [5] W.Ekardt,Phys.Rev.B3l (1985) 6360; D.E. Beck, Phys. Rev. B 35 (1987) 7325. 6] A.A. Lushnikov and A.J. Simonov, Z. Phys. 270 (1974)17; A.A. Lushnikov, V.V. Maksimenko and A.J. Simonov, in: Electromagnetic surface modes, ed. A.D. Boardman (Wiley, New York, 1982). [7] V. Kresin, Phys. Rev. B 38 (1988), to be published. [8] A.B. Migdal, Theory of flnite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967). [9] W.A. de Heer, W.D. Knight, M.Y. Chou and M.L. Cohen, in: Solid state physics, Vol. 40, eds. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1987).
PHYSICS LETTERS A
31 October 1988
COLLECI1VE RESONANCE FREQUENCIES IN METAL CLUSTERS Vitaly KR~SIN Department ~f Physics,
University of California, Berkeley, CA 94720, USA
Received 7 J~.i1y1988; revised manuscript received 24 August 1988; accepted for publication 25 August 1988 Communicaledby A.A. Maradudin
The collective resonance frequency of valence electrons in small metal clusters is calculated analytically. The self-consistent solution takds into account the density profile of the valence electrons, as given by the recently developed statistical approach. The resonance frequency is red shifted with respect to its classical value, and depends on the amount ofthe electron cloud’s spillout outside the boundary of the positive cluster core.
The visible spectrum of small metal particles and clusters is domjnated by the surface plasma resonance of the val~nceelectrons (see, e.g., refs. [1—3]). For metal spheres of radius much less than the wavelength of incide~itlight, the classical theory ,[4] predicts a resonance at the frequency w~= ~ where air, is the bulk plasma frequency. Various calculations have predicted both a red shift and a blue shift of the resonance compared to the classical value, depending on the ~nodelused (see e.g. the review [1], and the recent flumerical calculations [51). In recent experiment~on small sodium clusters [2], a red shift was obser%~ed. Here we present results of an analytical calculation of the surface i~esonancefrequencies, applicable to all small metal diusters in which optical properties in the visible rang~are dominated by delocalized valence electrons. We solve the integral equation for the resonance fleld [61 and utilize the recently developed Thoma~—Fermi(TF) statistical description of metal clusterS [7]. The position of the resonance is found to be 4etermined by the density profile of the valence ele~trons.The spill-out of the electron cloud outside flue boundary of the positive core of the cluster lead~to a red shift of the resonance; the magnitude of t~ieshift depends sensitively on the amount of the spill-out. The latter is calculated using the aforementioned statistical method, which describes the electron density distribution in clusters, This method has been successful in treating cluster
properties governed by the valence electrons, such as relative stabilities, ionization potentials and diamagnetism [7]. The general integral equation describing self-supporting oscillations in an electron gas in the randomphase approximation is (see, e.g., ref. [81) V(r) =e2
J
d3r 1 r— r1 I —
X
~
w~(r~ )~ti~(r1) ~ A A
A
V~ ,
(1)
A —
where V(r) is the field inside the system; ~A and P~VA are the single-particle energies and wave functions, respectively. The frequency w. for which a non-trivial solution exists is the resonance frequency. In ref. [61 it is shown that this equation can be brought into the following form: (1_ 4En(r) e~ V(r) \. mw~ ) 2 3r = — ci 1 V(r1 ) v1 1 mw~ Ir—r1 _-~_-~-
J
v1 n(r1) (2)
where n (r) is the number density of valence electrons in the particle. The authors substitute the stepfunction profile n (r) = nO(R — r) into eq. (2) and recover the classical surface plasmon frequency, w~.
0375-9601 1881$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
89
Volume 133, number 1,2
PHYSICS LETTERS A
For small clusters, it is essential to take into account the correct form of the electron density distribution n (r). For metal clusters, the TF statistical method [7] can be used to describe this quantity. Consider the case of spherical metal clusters (i.e. the “magic number” clusters with spherical closed shells [9]. The positive ion background of the cluster is treated as a uniform sphere of radius R = 3(a aor~N~/ 0 is the Bohr radius, r~is the bulk Wigner—Seitz density parameter and Na is the number of atoms in the cluster), The function n ( r) decreases rapidly in the region r~R [7], and its derivative is therefore sharply peaked in this region. This fact allows eq. (2) to be solved for ~ analytically. After some calculations, we find that the ratio of the resonance frequency to the classical surface plasmon frequency is given by 23g(R)+l
2(a)r/ws)
—{[3g(R)— 1 ]2+24q(l
—
3q)} 1/2,
(3)
where q=~Jg(r)dr,
k6
(r/R)
3/2
—
(r/R
—
D)
6,
(5)
where k=(l2/it)3r~”2N,Y3 D=l—
61 k
12 — —~--
k
+
...
,
(6)
and Ne is the number of valence electrons in the cluster. For a chosen cluster size and material, it is thus possible to calculate the resonance frequency. For instance, for Na 8, we find that the ratio ofthe latter to the classical surface plasmon frequency is WrI = 0.74; for Na20 it is 0.79, and for Na92, 0.84. (Recent experiments [2] have shown that for Na8 this
90
ratio is 0.75 ±0.02; our theory is in excellent agreement with these data.) To summarize, we have calculated the collective resonance frequencies of small metal clusters. This was done by solving the RPA integral equation [61 and accurately taking into account the density distribution of valence electrons in clusters. The results ofthe TF statistical method [7] are used for the density. The spill-out of the electron cloud inside the boundary of the positive background is shown to lead to a red shift in the position of the resonance, and the magnitude of the shift is directly related to the amount of the spill-out. This approach can be extended to calculate the static electric polarizability of metal clusters; this calculation will be presented elsewhere. I would like to thank Professor W.D. Knight, Dr. V.Z. Kresin and Kathy Selby for very useful discussions. This work was supported by the U.S. National Science Foundation under Grant No. DMR-8615246.
(4)
and g(r) is the dimensionless electron density profile defined by n(r)=p~g(r),p~being the number density of the uniform positive background charge. According to the TF solution [7], the function g( r) for r~R is given by g( r) =
31 October 1988
References [1] U. Kreibig and L. Genzel, Surf. Sci. 156 (1985) 678. [2] W.A. de Heer, K. Selby, V. Kresin, J. Masui, M. Vollmer, A. Châtelain and W.D. Knight, Phys. Rev. Lett. 59 (1987)1805; K. Selby, M. Voilmer, J. Masui, V. Kresin, W.A. de Heer and W.D. Knight, in: 4th mt. Symp. on Small particles and inorganic clusters, to be published in Z. Phys., K. Selby, M. Vollmer, J. Masui, V. Kresin, W.A. de Heer and W.D. Knight, to be published. [3] W. Hoheisel, K. Jungmann, M. Voilmer, R. Weidenauer and F.Trager,Phys.Rev.Lett.60(l988) 1649. [4] G.Mie,Ann.Phys.25 (1908) 377. [5] W.Ekardt,Phys.Rev.B3l (1985) 6360; D.E. Beck, Phys. Rev. B 35 (1987) 7325. 6] A.A. Lushnikov and A.J. Simonov, Z. Phys. 270 (1974)17; A.A. Lushnikov, V.V. Maksimenko and A.J. Simonov, in: Electromagnetic surface modes, ed. A.D. Boardman (Wiley, New York, 1982). [7] V. Kresin, Phys. Rev. B 38 (1988), to be published. [8] A.B. Migdal, Theory of flnite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967). [9] W.A. de Heer, W.D. Knight, M.Y. Chou and M.L. Cohen, in: Solid state physics, Vol. 40, eds. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1987).