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Theoretical analysis of the fluorescence yield of rare earth ions in glasses containing small metallic particles
Volume 174, number I
CHEMICAL PHYSICS LETTERS
26 October 1990
Theoretical analysis of the fluorescence yield of rare earth ions in glasses containing small metallic particles O.L. Malta Departamento de Quimica Fundamental, UFPE, Cidade Universit&ria,SO739Recife. PE, Brazil
and M.A. Couto dos Santos Dwrtamento de Fisica, UFPE, Cidade Universit&ria,50739 Reci&e,PE, Brazil Received I7 July 1990
We examine the fluorescence yield of rare earth ions in glasses containing a uniform volumetric distribution of metallic particles with small specific volume. The role of energy transfer between the ions and the particles is discussed. Depending on the particle size and the ion emission quantum efficiency, quenching or enhancement of the fluorescence is predicted.
1. Introduction
The observation of Raman scattering and fluorescence enhancements from molecules adsorbed or close to a rough metallic surface [ 1,2] has motivated the study of the optical properties of composite materials - fluoroborate glasses containing small silver particles - doped with the Eu3+ ion [ 3-61. In these systems, any significant influence of the metallic particles on the absorption and emission rates of the rare earth ions should be primarily of electromagnetic origin, though it is possible that chemical effects [ 7 ] are of importance to those ions situated on the particle surface. The electromagnetic mechanism can be viewed as an additional interaction due to the high field gradients, nearby the metallic particles, produced by plasmon excitations, in the particles, at the Mie resonance frequency. A basic interest in these composite materials is to see under what conditions the emission yield, which is a balance between emission and decay rates, can be optimized. This is in general a rather complex problem since it requires the knowledge of emission and decay rates, in the absence and in the presence of the metallic particles,
and particle concentration and size distribution as well. The decay rate of the emitting rare earth ion corresponding to a depopulating channel which is in resonance with the plasmon excitation can be greatly increased not only due to local field enhancement effects but also due to energy transfer from the ions to the metallic particles. (Energy transfer from the particles to the ions is not expected to be operative since the plasmon lifetime T, is extremely short, z lo-l4 s.) Therefore, if this is not compensated by the local field enhancement effects on the absorption rates, a quenching of the fluorescence yield may occur. Several theoretical approaches have been proposed to describe absorption and decay rates of molecular species in the presence of rough metallic surfaces and particles [ g-101. In this paper we present a theoretical treatment of the fluorescence yield of rare earth ions, in the presence of a uniform bulk distribution of metallic particles, as a function of particle- and host- (glass medium) dependent parameters. The role of energy transfer between the particles and ions, which was not explicitly included in a previous work [ 3 1, will be examined in terms of a quan-
0009-2614/90/S 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
13
Volume 174, number 1
CHEMICAL PHYSICS LETTERS
.tum mechanical expression which emphasizes the dependence of the transfer rate with the particle parameters and the energy mismatch condition.
26 October 1990
lP>
k3
I
G
TP
I2>
,2. Fluorescenceyield WPm
We consider a cubic arrangement of spherical metallic particles, with a sharp size distribution of average radius a, embeddedin a glass of dielectric constant codoped with rare earth ions. We also consider that the particles are separated by a distance 2Ro such that their specific volume q (proportional to a’/Ra) is small enough to allow the neglect of strong correlation between the particles (in general qSO.03). These features may be produced experimentally through the well known techniques of prep aration of these composite media [ 11,121. Indeed the interest in having rather small values of q lies in the fact that as the composite tends to a dense compact of metallic particles (large q) , their effective polarizabilities tend to decrease. This not only contributes negatively to the local field enhancement effects but also provides more favorable conditions to the energy transfer from the ions to the particles. Moreover, due to the large absorption coefficient of the metallic particles, the energy loss per incident radiation flux would be so high that the comparison between the fluorescence yields in the presence and in the absence of particles could be meaningless. As in ref. [ 3 1, our model consists in assuming each metallic particle concentric with a sphere of radius R. and neglecting the influence, over the rare earth ions inside this sphere, of the remaining particles. We then consider the level scheme shown in fig. 1 where [p > is the plasmon state, 7P is its lifetime and W, is the energy transfer rate for the pair ionparticle. In the case of the Eu3+ ion and silver particles (plasmon band at cz400 nm), for instance, W23 is an effective non-radiative rate from either a pure 4f level (the 5b, which is immediately above the 5D3) or a molecular-like level, depending on whether the host glass is transparent or not in the region of the plasmon excitation, to the fluorescing level 5DP In the absence of particles, the transition intensity from level 2 to level 1 is given by 14
wso
wi
WI0
II>
Im>1
L
lo>
RAREEARTHm
PLASMON
Fig. 1. Energy level scheme for a resonant plasmon and rare earth ion absorption.The double arrowindicatesthe fluorescent transition under consideration. Full arrows indicate radiative transitions, while the curved arrows indicate non-radiative transitions and the da&d arrow indicates non-radiative energy transfer.
112 =fw2
WnNr12
,
(1)
where N is the number of emitting ions, Aw is the transition energy and r12is the normalized population of level 2, which is obtained from the rate equations
dv2 -=-
;
q2 +
@%3tf3
-Q3 =- ; dt
f/3 +
W’Oll,
dt
(2)
and (3)
.
If the incident radiation is not intense enough such that the ground state IO} is very little depleted, then qox 1. Thus, in the steady state regime, dqJdt=O, we get from eqs. ( 1)- ( 3) 112=fio~i~Nw~ W23
W307273
-
(4)
In the presence of particles (indicated by the asterisk), since we will be interested in the evaluation of relative intensities, we restricts the analysis to a space of volume $cR i concentric with a particle. The high field gradients nearby the particle make the transition rates and lifetimes, related to the emitting ions, dependent on the particle-ion distance, R. This dependence is significant only for those channels
26October1990
CHEMICAL PHYSICS LETTERS
Volumef 74,numberI
which are in resonance with the plasmon excitation. Therefore, the rare earth ions are no longer equivalent and the intensity 1y, is now given by Ro
I;,=fi~,~4xC’&
(13) v;(R) RZ CM,
(5)
II
where
where C* is the concentration of ions. In the steady state regime the rate equations now give +)=W,,G(R)
3
(6)
rlf(R)=KoW+WR)vp
(7)
%= I+‘,,+ W,(R) G(R) 9
(8)
where we have considered v,,,x 1 and that the nonradiative rate W,, is not affected by the particles. Solving for q;(R) we get @(A)= W,,rT
’
Wfo(R)[W,(R) + l/al + wpmW,(R) [1/73(R)+W,(R)l[W,(R)+ll.rpl-W~(R)' (9)
The composite medium is characterized by a complex dielectric function (for small q) [31 E(W)=eo+
40; co;-w2-1yw'
(10)
where CO,is the metal plasma frequency, y is the damping factor and c+= [ ( 1-q)/3~o]1’2w,, is the Mie resonance frequency for the plasmon excitation in the particles. The appropriate forms of W&(R), 7$(R) and W,(R) have been given in refs. [3,13]. These are (11)
and
2 1 WP a=j~~-w2-iiy~
a3
(14)
is the polarizability of the metallic particles, p is the dipole matrix element of the rare earth transition which is in resonance with the plasmon excitation and d is the energy difference which gives the mismatch condition between donor (rare earth ion) and acceptor (particle) in the energy transfer process. Eqs. ( 11) and ( 12) deserve a comment. They have been derived on the basis of the pseudo multipolar field interaction which in the usual cases of 4f-4f transitions provides transition rates varying as combinations of R-8, R-l2 and R-l6 [3] where R, the rare earth ion-ligand distance, is typically of the order of 2.5 A. However, for large distances, Rr 10 A, its dipolar component by far dominates due to the action of the odd part of the ligand field Hamiltonian which relaxes Laporte’s rule. Thus, it may be shown that the square of a spherical component of the additional field produced by the metallic particle (taken on the z axis) is proportional to (3Q2-~)%I?/R~ (Q=O, kl), which, if one is not interested in polarization effects, may be replaced by the average value 2aZ/R6. The transition rates Wzo and W$, correspond to W,, and W,, (in the absence of particles) modified by the frequency dependent macroscopic Lorentz field correction. They are related by absorption,
x$9
emission ,
(15)
is the index of refraction of the where n*= Rem composite. We now make the following considerations. Since the transitions from level 2 are out of resonance with 15
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CHEMICAL PHYSICS LETTERS
the plasmon excitation, we may take W;, = W,, and 7; = 7z,. The plasmon lifetime z, is extremely short, x IO- I4 s, so that I /TV dominates all rates in fig. 1 and we can make in eq. (9) 1/7p~ WT. Noting that for the comparison between the fluorescence intensities in the presence and in the absence of particles the number of emitting ions in the volume $rRi must be the same, the ratio between concentrations is C*/C=Ri/(Ri-a3).Thus,fromeqs. (9), (5)and (4) we get the relative fluorescence yield
may take W,,l W,,x 106. These values presuppose that the transition IO) -f 13) is a pure 4f-4f transition, i.e. the host glass is transparent in the region of the plasmon excitation. The damping factor (the half-width of the plasmon absorption band), which is a function of the particle radius, may be given, according to the results of Kawabata and Kubo [ 16 1, by
y(a)=
(16) where M= W&l W,,z,, L=k+2a2M and Q=k+ 2a 2W& . This is similar to our previous result except that in eq. ( 16) energy transfer is taken into account.
3. Results In order to examine the behavior of eq. ( 16) with particle, ion and host dependent parameters, we will assume typical values for the transition rates and the dipole matrix element CL.Non-radiative transitions of rare earth ions in glasses, which are strongly dependent of the phonon density of states of the medium, may vary from x 1 ms to several p. Thus, we may take for W23 values in the range 103-lo5 s-l. From the theory of 4f-4f intensities [ 141, if level 3 in fig. 1 corresponds to the ‘Lb of the Eu3+ ion, with the S2,intensity parameter assuming the typical value of 10-20 cm’ and the matrix elements given in ref. [ 15] we obtain Wi3 x 1O2s-’ and $/e*z 1O-22cm2, where e is the electronic charge. By the same way, we get an oscillator strength for the 1O} -+ 13) transition of the order of 10m6.The plasmon band has the features of an allowed transition and, therefore, an oscillator strength of the order of 1, which can also be obtained from the relation between the plasma frequency o, and the average number of conduction electrons per unit volume in the particle. Then, we 16
26 October 1990
(
1.2+ +7
+10-7a3
>
x10’4s-i
)
(17)
where a is in A. In this way, mean free path effects are taken into account. With the values co=2.2S and, for silver, w,=1.35X 1016 s-t we have plotted I%/Z,, as function of the particle radius, at the Mie resonance frequency (w=wR=5X lOis s-i, f&O), for different values of W23 and Ro. The results are shown in figs. 2, 3 and 4. The curves were plotted for ranges of u values compatible with a ratio a’/Ri BO.03. Extrapolations could be done, though, for the reasons commented above, the results would be less sure to be compared with experiment. The first point that can be noted is the increase in the relative fluorescence yield with the non-radiative rate W,,.This means that
IO
8
s-46
I-i.
*TV
-4
2
Fig. 2. Fluorcsccnce gain as a function of particle radius for R,,= 150 A and Wz3= 10) s-l (curve l), IO4s-l (curve 2) and 10’ s-’ (curve 3).
Volume 174. number 1
CHEMICAL PHYSICS LETTERS
26 October 1990
particle radii a fluorescence quenching occurs (&/IIZ < 1). This quenching effect would be attenuated for a plasmon peak excitation out of resonance (df 0) with level 3, provided that the plasmon width is such that the energy mismatch is non-vanishing.
4. Concludingremarks
Fig. 3. The same as in fig. 2 for R,= 300A.
50
40
NX z . *N -2c
IC
Fig. 4. The same as in figs. 2 and 3 for Ro=500A.
a fluorescence enhancement is favorized in ions with high emission quantum efficiencies. This is easily understandable since W,, corresponds to the populating channel of the fIuorescing state 12). The competition between the local field enhancement and energy transfer effects can also be noted, particularly in fig. 2. This is the reason for the minima in the curves. For low W,, values and smaller
Although we have specifically considered the Eu3+ ion, the energy diagram in fig. 1 may be applied to other rare ions as well (Dy3+, Ho3+ and Er3+ are good candidates), and since the values used for the transition rates are very typical of 4f-4f transitions, the results shown in figs. 2, 3 and 4 have actually a more general character. They predict that, with small q values, considerable fluorescence enhancements may be obtained for ions with high emission quantum efficiencies depending on the particle size. From the other side, for smaller quantum efficiencies a fluorescence quenching may occur due to energy transfer from the ions to the particles. In our model disorder and particle coagulation effects were not considered. We believe, however, that here, for small specific volumes, these effects are of much less importance than in the case of molecular species deposited on rough metallic surfaces [ 171. In a previous experimental result on the fluorescence enhancement of the Eu3+ ion (an observed gain of ~6) [ 31, the plasmon band has been assigned at x 314 nm, in resonance with the highest level of the 4f6 configuration, where the fluoroborate host glass starts absorbing Indeed this band may be due to very fine silver particles with radii from less than 10 to 20 8, [ 181. There may be also a broadening of the absorption edge of the host glass due to Ag+ ions and Ag atoms [ 5,6]. In this case energy transfer from the host medium to the Eu3+ ions would be quite efficient, also contributing to the fluorescence enhancement. In ref. [ 3 1, the fluorescence gain was interpreted in terms of the local field enhancement effects, and since energy transfer from the Eu3+ ions to the silver particles was not taken into account it is quite possible that the effect of broadening of the host absorption edge due to Ag+ ions and Ag atoms was operative in that case. A point concerning the preparation of these composite media should be mentioned. The shape, size 17
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CHEMICAL PHYSICS LETTERS
and distribution of metallic particles in glasses are quite sensitive to the method of nucleation and growth used. There are methods by which these parameters can be controlled. Nevertheless, there are evidences that the particles may not be purely metallic [ 191. This would introduce modifications in the particle polarizability and in the energy transfer rate, W,, which cannot be readily evaluated. It is likely that this composition effect is of more relevance for the smaller particles.
Acknowledgement The authors are grateful to the CNPq and CAPES (Brazilian Agencies) for financial support.
References [l] M. Fleischman, P.J. Hendra and A.J. McQuillan, Chem. Phjis. Letters 26 ( 1974) 123. [2] D.A. Weitz, S. Garoff, C.D. Hanson, T.J. Gramila and J. Gersten, J. Luminescence 24/25 (1981) 83. [3] O.L. Malta, P.A. Santa-&z, G.F. de %I and F. Auzel, J. Luminescence 33 (1985) 261.
18
26 October 1990
[4] O.L. Malta, P.A. Santa-Cruz, G.F. de !%and F. Auzel, Chem. Phys. Letters 116 (1985) 396. [5] G.F. de S& W.M. de AzevMo and O.L. Malta, J. Luminescence 40/41 ( I98 1) 133. [6]W.M,deAzevtio,O.L.Malta.,G.EdeSgandLP.Padovan, QuIm. Nova 11 (1988) 96. [7] C. Pettenkofer, I. Mrozek, T. Bomemann and A. Otto, Surface Sci. 188 (1987) 519. [S] D.A. Weitz, S. Garoff, J-1.Gersten and A. Nitzan, J. Chem. Phys. 78 ( 1983) 5324. [ 91 P. Das and H. Metiu, J. Phys. Chem. 89 ( 1985) 4680. [lO]H.Chew,J.Chem.Phys.87 (1987) 1355. [ 111W.J. Kaiser, E.M. Logothetis and L.E. Wenger, J. Phys. C 18 (1985) L837. [ 121Sung-Ik Lee, T.W. Noh, J.R. Gaines, Ying-Hsiang Ko and E.R. Kreidler, Phys. Rev. B 37 (1988) 2918. [13]0.L.Malta,Phys.LettersA114(1986) 195. [ 141RD. Peacock, Struct. Bonding 22 (1975) 83. [ 151W.T. Camall, H. Crosswhite and H.M. Crosswhite, Energy level structure and transition probabilities of the trivalent lanthanides in LaF (Argonne National Laboratory, Argonne) [ 161A. Kawabata and R. Kubo, J. Phys. Sot. Japan 21 ( 1966) 1765. [ 171N. Liver, A. Nitzan and J.I. Gersten, Chem. Phys. Letters 111 (1984) 449. [ IS] L. Genzel, T.P. Martin and U. Kreibig, Z. Physik 2 1 ( 1975) 339.
[ 191 G.F. de S& O.L. Malta, W.M. de Azevedo and H. Dexpert, J. Less-Comm. Met. 148 (1989) 387.
CHEMICAL PHYSICS LETTERS
26 October 1990
Theoretical analysis of the fluorescence yield of rare earth ions in glasses containing small metallic particles O.L. Malta Departamento de Quimica Fundamental, UFPE, Cidade Universit&ria,SO739Recife. PE, Brazil
and M.A. Couto dos Santos Dwrtamento de Fisica, UFPE, Cidade Universit&ria,50739 Reci&e,PE, Brazil Received I7 July 1990
We examine the fluorescence yield of rare earth ions in glasses containing a uniform volumetric distribution of metallic particles with small specific volume. The role of energy transfer between the ions and the particles is discussed. Depending on the particle size and the ion emission quantum efficiency, quenching or enhancement of the fluorescence is predicted.
1. Introduction
The observation of Raman scattering and fluorescence enhancements from molecules adsorbed or close to a rough metallic surface [ 1,2] has motivated the study of the optical properties of composite materials - fluoroborate glasses containing small silver particles - doped with the Eu3+ ion [ 3-61. In these systems, any significant influence of the metallic particles on the absorption and emission rates of the rare earth ions should be primarily of electromagnetic origin, though it is possible that chemical effects [ 7 ] are of importance to those ions situated on the particle surface. The electromagnetic mechanism can be viewed as an additional interaction due to the high field gradients, nearby the metallic particles, produced by plasmon excitations, in the particles, at the Mie resonance frequency. A basic interest in these composite materials is to see under what conditions the emission yield, which is a balance between emission and decay rates, can be optimized. This is in general a rather complex problem since it requires the knowledge of emission and decay rates, in the absence and in the presence of the metallic particles,
and particle concentration and size distribution as well. The decay rate of the emitting rare earth ion corresponding to a depopulating channel which is in resonance with the plasmon excitation can be greatly increased not only due to local field enhancement effects but also due to energy transfer from the ions to the metallic particles. (Energy transfer from the particles to the ions is not expected to be operative since the plasmon lifetime T, is extremely short, z lo-l4 s.) Therefore, if this is not compensated by the local field enhancement effects on the absorption rates, a quenching of the fluorescence yield may occur. Several theoretical approaches have been proposed to describe absorption and decay rates of molecular species in the presence of rough metallic surfaces and particles [ g-101. In this paper we present a theoretical treatment of the fluorescence yield of rare earth ions, in the presence of a uniform bulk distribution of metallic particles, as a function of particle- and host- (glass medium) dependent parameters. The role of energy transfer between the particles and ions, which was not explicitly included in a previous work [ 3 1, will be examined in terms of a quan-
0009-2614/90/S 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
13
Volume 174, number 1
CHEMICAL PHYSICS LETTERS
.tum mechanical expression which emphasizes the dependence of the transfer rate with the particle parameters and the energy mismatch condition.
26 October 1990
lP>
k3
I
G
TP
I2>
,2. Fluorescenceyield WPm
We consider a cubic arrangement of spherical metallic particles, with a sharp size distribution of average radius a, embeddedin a glass of dielectric constant codoped with rare earth ions. We also consider that the particles are separated by a distance 2Ro such that their specific volume q (proportional to a’/Ra) is small enough to allow the neglect of strong correlation between the particles (in general qSO.03). These features may be produced experimentally through the well known techniques of prep aration of these composite media [ 11,121. Indeed the interest in having rather small values of q lies in the fact that as the composite tends to a dense compact of metallic particles (large q) , their effective polarizabilities tend to decrease. This not only contributes negatively to the local field enhancement effects but also provides more favorable conditions to the energy transfer from the ions to the particles. Moreover, due to the large absorption coefficient of the metallic particles, the energy loss per incident radiation flux would be so high that the comparison between the fluorescence yields in the presence and in the absence of particles could be meaningless. As in ref. [ 3 1, our model consists in assuming each metallic particle concentric with a sphere of radius R. and neglecting the influence, over the rare earth ions inside this sphere, of the remaining particles. We then consider the level scheme shown in fig. 1 where [p > is the plasmon state, 7P is its lifetime and W, is the energy transfer rate for the pair ionparticle. In the case of the Eu3+ ion and silver particles (plasmon band at cz400 nm), for instance, W23 is an effective non-radiative rate from either a pure 4f level (the 5b, which is immediately above the 5D3) or a molecular-like level, depending on whether the host glass is transparent or not in the region of the plasmon excitation, to the fluorescing level 5DP In the absence of particles, the transition intensity from level 2 to level 1 is given by 14
wso
wi
WI0
II>
Im>1
L
lo>
RAREEARTHm
PLASMON
Fig. 1. Energy level scheme for a resonant plasmon and rare earth ion absorption.The double arrowindicatesthe fluorescent transition under consideration. Full arrows indicate radiative transitions, while the curved arrows indicate non-radiative transitions and the da&d arrow indicates non-radiative energy transfer.
112 =fw2
WnNr12
,
(1)
where N is the number of emitting ions, Aw is the transition energy and r12is the normalized population of level 2, which is obtained from the rate equations
dv2 -=-
;
q2 +
@%3tf3
-Q3 =- ; dt
f/3 +
W’Oll,
dt
(2)
and (3)
.
If the incident radiation is not intense enough such that the ground state IO} is very little depleted, then qox 1. Thus, in the steady state regime, dqJdt=O, we get from eqs. ( 1)- ( 3) 112=fio~i~Nw~ W23
W307273
-
(4)
In the presence of particles (indicated by the asterisk), since we will be interested in the evaluation of relative intensities, we restricts the analysis to a space of volume $cR i concentric with a particle. The high field gradients nearby the particle make the transition rates and lifetimes, related to the emitting ions, dependent on the particle-ion distance, R. This dependence is significant only for those channels
26October1990
CHEMICAL PHYSICS LETTERS
Volumef 74,numberI
which are in resonance with the plasmon excitation. Therefore, the rare earth ions are no longer equivalent and the intensity 1y, is now given by Ro
I;,=fi~,~4xC’&
(13) v;(R) RZ CM,
(5)
II
where
where C* is the concentration of ions. In the steady state regime the rate equations now give +)=W,,G(R)
3
(6)
rlf(R)=KoW+WR)vp
(7)
%= I+‘,,+ W,(R) G(R) 9
(8)
where we have considered v,,,x 1 and that the nonradiative rate W,, is not affected by the particles. Solving for q;(R) we get @(A)= W,,rT
’
Wfo(R)[W,(R) + l/al + wpmW,(R) [1/73(R)+W,(R)l[W,(R)+ll.rpl-W~(R)' (9)
The composite medium is characterized by a complex dielectric function (for small q) [31 E(W)=eo+
40; co;-w2-1yw'
(10)
where CO,is the metal plasma frequency, y is the damping factor and c+= [ ( 1-q)/3~o]1’2w,, is the Mie resonance frequency for the plasmon excitation in the particles. The appropriate forms of W&(R), 7$(R) and W,(R) have been given in refs. [3,13]. These are (11)
and
2 1 WP a=j~~-w2-iiy~
a3
(14)
is the polarizability of the metallic particles, p is the dipole matrix element of the rare earth transition which is in resonance with the plasmon excitation and d is the energy difference which gives the mismatch condition between donor (rare earth ion) and acceptor (particle) in the energy transfer process. Eqs. ( 11) and ( 12) deserve a comment. They have been derived on the basis of the pseudo multipolar field interaction which in the usual cases of 4f-4f transitions provides transition rates varying as combinations of R-8, R-l2 and R-l6 [3] where R, the rare earth ion-ligand distance, is typically of the order of 2.5 A. However, for large distances, Rr 10 A, its dipolar component by far dominates due to the action of the odd part of the ligand field Hamiltonian which relaxes Laporte’s rule. Thus, it may be shown that the square of a spherical component of the additional field produced by the metallic particle (taken on the z axis) is proportional to (3Q2-~)%I?/R~ (Q=O, kl), which, if one is not interested in polarization effects, may be replaced by the average value 2aZ/R6. The transition rates Wzo and W$, correspond to W,, and W,, (in the absence of particles) modified by the frequency dependent macroscopic Lorentz field correction. They are related by absorption,
x$9
emission ,
(15)
is the index of refraction of the where n*= Rem composite. We now make the following considerations. Since the transitions from level 2 are out of resonance with 15
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CHEMICAL PHYSICS LETTERS
the plasmon excitation, we may take W;, = W,, and 7; = 7z,. The plasmon lifetime z, is extremely short, x IO- I4 s, so that I /TV dominates all rates in fig. 1 and we can make in eq. (9) 1/7p~ WT. Noting that for the comparison between the fluorescence intensities in the presence and in the absence of particles the number of emitting ions in the volume $rRi must be the same, the ratio between concentrations is C*/C=Ri/(Ri-a3).Thus,fromeqs. (9), (5)and (4) we get the relative fluorescence yield
may take W,,l W,,x 106. These values presuppose that the transition IO) -f 13) is a pure 4f-4f transition, i.e. the host glass is transparent in the region of the plasmon excitation. The damping factor (the half-width of the plasmon absorption band), which is a function of the particle radius, may be given, according to the results of Kawabata and Kubo [ 16 1, by
y(a)=
(16) where M= W&l W,,z,, L=k+2a2M and Q=k+ 2a 2W& . This is similar to our previous result except that in eq. ( 16) energy transfer is taken into account.
3. Results In order to examine the behavior of eq. ( 16) with particle, ion and host dependent parameters, we will assume typical values for the transition rates and the dipole matrix element CL.Non-radiative transitions of rare earth ions in glasses, which are strongly dependent of the phonon density of states of the medium, may vary from x 1 ms to several p. Thus, we may take for W23 values in the range 103-lo5 s-l. From the theory of 4f-4f intensities [ 141, if level 3 in fig. 1 corresponds to the ‘Lb of the Eu3+ ion, with the S2,intensity parameter assuming the typical value of 10-20 cm’ and the matrix elements given in ref. [ 15] we obtain Wi3 x 1O2s-’ and $/e*z 1O-22cm2, where e is the electronic charge. By the same way, we get an oscillator strength for the 1O} -+ 13) transition of the order of 10m6.The plasmon band has the features of an allowed transition and, therefore, an oscillator strength of the order of 1, which can also be obtained from the relation between the plasma frequency o, and the average number of conduction electrons per unit volume in the particle. Then, we 16
26 October 1990
(
1.2+ +7
+10-7a3
>
x10’4s-i
)
(17)
where a is in A. In this way, mean free path effects are taken into account. With the values co=2.2S and, for silver, w,=1.35X 1016 s-t we have plotted I%/Z,, as function of the particle radius, at the Mie resonance frequency (w=wR=5X lOis s-i, f&O), for different values of W23 and Ro. The results are shown in figs. 2, 3 and 4. The curves were plotted for ranges of u values compatible with a ratio a’/Ri BO.03. Extrapolations could be done, though, for the reasons commented above, the results would be less sure to be compared with experiment. The first point that can be noted is the increase in the relative fluorescence yield with the non-radiative rate W,,.This means that
IO
8
s-46
I-i.
*TV
-4
2
Fig. 2. Fluorcsccnce gain as a function of particle radius for R,,= 150 A and Wz3= 10) s-l (curve l), IO4s-l (curve 2) and 10’ s-’ (curve 3).
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26 October 1990
particle radii a fluorescence quenching occurs (&/IIZ < 1). This quenching effect would be attenuated for a plasmon peak excitation out of resonance (df 0) with level 3, provided that the plasmon width is such that the energy mismatch is non-vanishing.
4. Concludingremarks
Fig. 3. The same as in fig. 2 for R,= 300A.
50
40
NX z . *N -2c
IC
Fig. 4. The same as in figs. 2 and 3 for Ro=500A.
a fluorescence enhancement is favorized in ions with high emission quantum efficiencies. This is easily understandable since W,, corresponds to the populating channel of the fIuorescing state 12). The competition between the local field enhancement and energy transfer effects can also be noted, particularly in fig. 2. This is the reason for the minima in the curves. For low W,, values and smaller
Although we have specifically considered the Eu3+ ion, the energy diagram in fig. 1 may be applied to other rare ions as well (Dy3+, Ho3+ and Er3+ are good candidates), and since the values used for the transition rates are very typical of 4f-4f transitions, the results shown in figs. 2, 3 and 4 have actually a more general character. They predict that, with small q values, considerable fluorescence enhancements may be obtained for ions with high emission quantum efficiencies depending on the particle size. From the other side, for smaller quantum efficiencies a fluorescence quenching may occur due to energy transfer from the ions to the particles. In our model disorder and particle coagulation effects were not considered. We believe, however, that here, for small specific volumes, these effects are of much less importance than in the case of molecular species deposited on rough metallic surfaces [ 171. In a previous experimental result on the fluorescence enhancement of the Eu3+ ion (an observed gain of ~6) [ 31, the plasmon band has been assigned at x 314 nm, in resonance with the highest level of the 4f6 configuration, where the fluoroborate host glass starts absorbing Indeed this band may be due to very fine silver particles with radii from less than 10 to 20 8, [ 181. There may be also a broadening of the absorption edge of the host glass due to Ag+ ions and Ag atoms [ 5,6]. In this case energy transfer from the host medium to the Eu3+ ions would be quite efficient, also contributing to the fluorescence enhancement. In ref. [ 3 1, the fluorescence gain was interpreted in terms of the local field enhancement effects, and since energy transfer from the Eu3+ ions to the silver particles was not taken into account it is quite possible that the effect of broadening of the host absorption edge due to Ag+ ions and Ag atoms was operative in that case. A point concerning the preparation of these composite media should be mentioned. The shape, size 17
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CHEMICAL PHYSICS LETTERS
and distribution of metallic particles in glasses are quite sensitive to the method of nucleation and growth used. There are methods by which these parameters can be controlled. Nevertheless, there are evidences that the particles may not be purely metallic [ 191. This would introduce modifications in the particle polarizability and in the energy transfer rate, W,, which cannot be readily evaluated. It is likely that this composition effect is of more relevance for the smaller particles.
Acknowledgement The authors are grateful to the CNPq and CAPES (Brazilian Agencies) for financial support.
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