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Effect of back-transfer on the energy transfer in Tb3+-doped glasses
= I~,
ELSEVIER
JOURNAL OF
LUMINESCENCE Journal of Luminescence 62 (1994) 69—76
Effect of back-transfer on the energy transfer in Tb3 + -doped glasses Kazuhiko Tonookaa, ~, Fumio Maruyama” 1, Norihiko Kamata”, Koji Yamada”, Jun Ono” Hokkaido National Industrial Research Institute, Sapporo 062, Japan b
Department of Functional Materials Science and Engineering, Saitama University, Urawa 338, Japan Received 8 December 1992; revised 18 April 1993, 8 February 1994; accepted 21 February 1994
Abstract The transient response of donor fluorescence under the influence of energy transfer has been analyzed using a non-linear model including the back-transfer effect ofenergy from acceptors to donors. In addition to the non-linear product of donor and acceptor populations and their improved distribution functions, the back-transfer process turned out to be important in describing the interacting donor—acceptor system. It was revealed th~tthe back-transfer ofenergy enhances the non-exponentiality of decay curves. The numerical analysis of energy transfer in Tb3 + -doped glasses showed a good agreement between experimental results and the model, which enables us to discuss the more detailed mechanism of energy transfer.
1. Introduction Energy transfer of electronic excitation from one ion (energy donor) to another (energy acceptor) has been attracting attention both for theoretical interest and for practical use such as phosphors and lasers. Since the fundamental works by Förster [1] and Dexter [2], various models for the energy transfer taking into account multi-polar interaction [2,3], exchange interaction [2,3], diffusion of energy [4,5] and the migration of energy [5,6], have been developed. A detailed bibliography on this field was prepared at the Los Alamos National Laboratory [7].
1
On leave from; Sumita Optical Mfg. Co., Ltd.
*
Corresponding author,
Förster analyzed decay curves of donor fluorescence by a linear rate equation of the donor population, disregarding whet~ieracceptors were excited or not. However, the population of acceptors is expected to play an important role to the energy transfer process with the increase of acceptor concentration. There are several possible models, such as the Inokuti—Hirayam~model [3], the cascade model [8], the discrete shell model [9] and the sum-of-exponentials model [10], for the non-exponential decay curves of donor fluorescence in particular conditions, although none of them give a general solution for au cases. There still remain theoretical problems and discrepancies between models and experimental results. The main difficulties of the Förster and Inokuti—Hirayama models~are the infinite energy transfer rate, which appears at the initial point of
0022-2313/94/$07.0O © 1994 — Elsevier Science By. All rights reserved SSDI 0022-2313(94)00019-9
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K. Tonooka et a!.
/ Journal of Luminescence 62
decay curves, and the completely random distribution of acceptors, even within the ionic radius, around a donor. The modification of acceptor distribution was already suggested as an approach to form a realistic distribution of active ions by Dornauf et al. [11]. In a previous paper [12], we proposed a non-linear model in which acceptors were assumed to be distributed outside the nearest neighbor distance of the donor and the macroscopic transfer rate of energy was finite at all time. Recently, we have found that the non-linear effect is one of the reasonable origins causing nonexponential responses under the influence energy transfer [13]. The differences between the non-linear model and the Förster model are outlined in the appendix. Numerical calculations showed that the non-linear model gives non-exponential decay curves similar to the Inokuti—Hirayama equation and a good fit to the experimental results of donor fluorescence, although the estimated energy relaxation rate of acceptors was much lower than expected. While there have been many reports on energy transfer between various rare-earth ions, only a few reports dealt with the backward transfer of electronic excitation from acceptors to donors (backtransfer) [14,15]. From the basic formula by Forster and Dexter, the energy transfer rate strongly depends on the overlap of emission and absorption spectra. Since the small Stokes shift of rare-earth ions and the inhomogeneous spectral broadening characteristic to glasses increase the magnitude of the overlap integral, the back-transfer, in addition to the forward-transfer, should be considered in the analysis of energy transfer. In this study, we extended the non-linear model by including backtransfer of energy and analyzed the fluorescence of Tb3 + ions in glasses. A better agreement between experiment and theory was obtained and the value of energy relaxation rate of acceptors became reasonable when compared with the multi-phonon relaxation rate.
2. Formulation We start from the following rate equation [12] for the population of donors and acceptors in the
(1994) 69—76
excited state, pd(t) and pa(t), respectively, under the influence of energy transfer,
dpd(t) dt
=
(1 —
—
Td’.
dPa(t)
—
(kR ~
—
k ‘1
“Tak’
—
—
kN)pd(t)
+
1
t
Pa’. 1/Pd Pa~.tJ)Pd~t) \ — “APa t
,
at where g is the excitation rate of donors, kR the radiative transition rate of donors, kN the nonradiative transition rate of donors, kA the energy relaxation rate of acceptors, kTd the energy transfer rate from a donor to acceptors, kTa the energy transfer rate from donors to an acceptor. In this paper, we extend the above rate equation by including the back-transfer of energy. One of the main factors affecting transfer rates is the overlap integral of emission and absorption spectra as derived by Förster and Dexter. Since the Stokes shift of rate-earth ions is small, the increased overlap integral will lead to an enhancement of the backtransfer. Furthermore, the phonon side band at room temperature [9] and the spectral broadening characteristic of glasses increase the transfer rate of both forward and backward directions. As a consequence, not only the forward-transfer but also the back-transfer play an important role in the energy transfer process. The Stokes shift of Tb3 + is considered to be same as the half width of the emission line shape from experimental results [16,17], as shown schematically in Fig. 1. They are typically about 10 nm in a silicate glass, therefore we can estimate the efficiency of the back-transfer as ~ = 0.2, assuming that the normalized emission spectrum of donors and the normalized absorption spectrum of acceptors are equal and that these line shapes of emission and absorption are Gaussian, as shown in the figure. The back-transfer rate of energy will be proportional to the acceptor population in excited state and the donor population in ground state. Thus, we obtained the rate equation for the population of donors and acceptors including the effect of back-transfer,
K. Tonooka et a!. / Journal of Luminescence 62 (1994) 69—76
0.12
71
_______________
F~=Absorptionof donor ~d=Emission of donQr
0
0.1 0
~d=
—
F
F 5=Absorption of acceptor ta ~Emission of acc8ptor •~ =Overlapof ~°~P°~ ~a8fldFd ~dafldFa
I’
°
a, I
0.08-
Fd/
CI)
~
~
/
0.06-
o
__________
I
0.04
/
-
o
/ I
cr0.02-
I S Fa(?~)dAI S fa(~FdQ’)d)~~O.O1~1 td(
/
z
_________________ +0.0681
/
oCl)
/ / /
0.00 350
360
S..
370
380
390
400
41
420
WAVELENGTH (nm) Fig. 1. Schematic spectra ofabsorptionand emission for donors and acceptors. The forward and back4ransfer rates are proportional to ~J~(a) Faa) d)~and Jf~(2)Fd(~.)dA, respectively. The back-transfer efficiency ‘la = 0.2 is estimated from these overlap integrals, by using a typical data of Tb’ + in glasses [16,17].
dpd(t) dt
=
(1 —
—
pd(t))g
kTd(l
—
—
Radiative Relaxation
(kR + kN)pd(t)
~kR
Pa(t))Pd(t) Forward-Transfer
+ flBkTdPa(t)(l
dpa(t)
dt
—
kTa(1
—
—
Pd(t)),
(2)
g
Pa(t))Pd(t)
—
—
?lBkTaPa(t)(l
—
Excitation
kid
Dono~
I
1B~d~
kia
Acceptors
BkTa
Back-Transfer
Pd(t)) — kAPa(t).
Radiativ kA~
e and Non-radiative
The macroscopic flow of energy is illustrated in Fig. 2. The intensity of the donor fluorescence is proportional to the product of pd(t) and the donor concentration Cd, and is expressed as 1(t)
it
Cdpd(t),
(3)
where it is a coefficient. It is possible to analyze the donor fluorescence 5D 5D using Eqs. (2) and (3). the itcase of 3 -+to set 4 3 +Inions, is reasonable cross-relaxation of Tb donors and acceptors are kTd = kTa = kT because Tb3 + ions. Since the 5D 4 state is the metastable
Non-radiative Relaxation
Relaxation
Fig. 2. A macroscopic model Øf excitation, relaxation, forwardtransfer and back-transfer of ~nergy. A donor gives energy to nearby acceptors and an acceJ~torobtains energy from nearby donors. Since the spatial distri~,utionsof donors and acceptors are generally different, these tr4nsfer rates kTd and k 1~should be distinguished in the analysis.
state of inthethedonor, the I~’ig. ~D43 state be considered analysis. showsshould the schematic flow of energy for the Tb~~—Tb3+ cross-relaxation. In the forward-transfer process, the donor relaxes
K. Tonooka et a!. / Journal of Luminescence 62 (1994) 69—76
72
where k4 is the energy relaxation rate of the 5D4 state. In order to analyze the microscopic proper-
Radiative Relaxation
5D I ExcitatIon
g
at
~+
3 state
d~
P
ties of energy transfer, we use following relations [12],
donors)
~(excited
k1
p Non-radiatIve Relaxation
=
ItDkR
~
+ 4irDk~fl~-~:
(5)
1~4
krII~11eki
and
~ij3
Tb3~at7F
Forward-Transfer ““~
at 5D 4 state
0state
D
=
4c4 3h 44
_____
2C )4
$
t~7a(E)
dE
ç fd(E)1’a(E)
dE,
(K !/
(excited acceptors)
(meta-stable donors)
(6) 3 ions and where R~ is the Cdminimum is the concentration distance between of Tb Tb3 ions, /3 is
,j,kA
+
~1(4 Radiative and Non-radiative
Radiative Relaxation and Non-radiative
+
Relaxation
+
Fig. 3. Schematic flow of ener~ifor the Tb’~—Tb’~cross-relaxation considering the metastable state. In the forward-transfer process, the donor relaxes from its excited state to its metastable state, giving a part ofexcitation energy to a ground state acceptor. In the back-transfer process, the energy of the acceptor transfers back to the metastable donor and thereby excites the donor to the excited state.
from its excited state to its meastable state, giving a part of excitation energy to a ground state acceptor. As a result of the cross-relaxation, the metastable donor and the excited acceptor appear enough in the distance to interact. In the backtransfer process, the energy of the acceptor transfers back to the metastable donor and thereby excites the donor to the excited state. Then 5Dthe rate equation including the population of 4 state p4(t) is given by dpd(t) dt
=
(1 —
pd(t))g
—
k1(1
—
—
(kR +
kN)pd(t)
Pa(t))Pd(t)
+ flBkTPa(t)P4(t),
dPa(t)
dt
=
4(t) dt
kT(l
—
(4)
Pa(t))Pd(t)
1CTPa(t)P4(t) — kAPa(t),
dp
—
=
1B
kT(1 —
a parameter characterizing the segregation of Tb3 ions, n is the refractive index, c is the velocity of light, K is the dielectric constant, e~is the permittiv-
—
Pa(t))Pd(t)
~iB kTP4(t)Pa(t)
—
k4p4(t),
ity of
medium, E is photon energy, dE) is the energy integral of the optical cross-section 7a(E) in the absorption band, fd(E) is the normalized emission spectrum of the donor and Fa(E) is the normalized absorption specQa( =
the
S 7a(E)
trum of the acceptor. We can set the quantity ~/, i/2e 1, equal to unity in the numerical evaluation. The interaction between donors and acceptors is assumed to be electric dipole—dipole in the following analyses for simplicity. In this case, the critical transfer distance R0 is equal to D 1/6
3. Experimental The glass samples under investigation contained Tb2O3 and Na20 in addition to the network formers (NWF) such as Si02, Ge02, Be2 03 and P2 0~.Their compositions are expressed in terms of molar percentages x, y and z as x NWF + y Na2O + zTb2O3, where x + y + z = 100 and y/x = 0.5. The glass samples with z = 0.3 to 7.0 for each NWF were prepared [12] as listed in Table 1. Before measuring the fluorescence intensity, all of the same glasses surface were cut condition. and polished The time in order dependence to ensure of 34 fluorescence intensity from the 5D 5D Tb 3 excited donor state and from the 4 metastable state in silicate glasses were simultaneously measured by the N2 laser (2 = 337 nm) pulse excitation, with the
K. Tonooka et a!.
/
Journal of Luminescence 62 (1994) 69—76
73
Table 1 Sample data of Tb’k-doped glasses NWF
SiO,
Ge0
Base glass composition [mol.%] Sample composition [mol.%] Doped Tb,O, [mol.%] Melting temperature Annealing ( — 20°C/h) Size
67SiO2 + 33Na,O 67GeO, + 33Na,O x NWF + y Na20 + z Tb,O, (x + y + z = z = 0.3, 0.7, 1.0, 1.5, 2.5, 4.0, 6.0 and 7.0 1300°C 1000°C 400—100°C 450—100°C 5 mm x 5 mm x 8 mm
2
1-
-.-
:~ I—
-~ >-
P205
67B2O, + 33’Na20 100, y/x = 0.5)
67P20, + 33Na2O
1200°C 450—100°C
900°C 200—100°C
5D 7F Tb’°( 3-+ 5)
Tb20, 0.3 mol% -~
B20,
~\
So’d curves wIth beck-transfer Dashed curvee:without beck-transfer Dofeexperlnrenbe
9
8 6 4
Z
uJ I—
\5..
z
7~.
Z
w
r5
w 0 z w ()
—
N
N
to w
~
cc O 0.1 -J
~.
U~
~
-
z
7F
cc 0 0.1 e W
°‘~
U) w
‘D—s
05
“1w
._J
Ô.0
I 0.5
I 1.0
I 1.5
4
t
~~ 2.0 20
TIME(msec)
\\
‘~°~
~ 4.Omoi% 50
2.5mol% ~
‘,
\
‘.1
~-.-
100
I 150
200
Fig. 4. Fluorescence decay curves of the 5D, —v ‘F 5 and the and glass. the slow of ‘D4 —v ‘F5 transitions of5D, Tb’fluorescence + in a silicate Therise nonexponential 5D decay of the the 4 fluorescence reflect the energy transfer between the ‘D4 (donor) and the ‘F6 —v ‘F0 (acceptor) transitions.
half width of 1 ns, as shown in Fig. 4. The non5D exponential 5D decay ofthe 3 emission and the slow rise of5D 4 emission reflect 5D the population 5D 5D 7F change 7F from 3 to 4 due to the 3 —‘ 4: 6 0 cross-relaxation, 3 + fluorescence in The decay Tb silicate glassescurves under for thethe influence of cross-relax—~
ation were analyzed numerically using the rate equation (4). All measurements were carried out at room temperature. 5D Fig. 5 shows the experimental points for the 3 ~‘F5emission and the fitted curves obtained by the non-linear analysis including back-transfer. Curve fitting parameters are —
TIME
( iisec
Fig. 5. The ‘D 3 —p ‘F5 (donor) decay curves analyzed using the non-linear model. The theor~tical curves with back-transfer gives better fit than those without back-transfer.
listed in Table = 1000 s and 1 from2.theWe 5Dput kR5D k4 = 250 s 3 and 4 decay curves for 0.3 mol.% of Tb2 0 A distance R0 of 0.4 nm was used,3~ which nitinimum distance between ions isintheTb two Tb rate kN was t~eglectedagainst 203. The non-radiative transition kR. The critical transfer distance R 0 is evaluated instead of the parameter D, assun~ingelectric dipole—dipole interaction. 5D The rapid decay of the fluorescence from the 3 state with increasing Tb2 03 content is due to the cross-relaxi~tion, 5Dwhich transfers electrons from the excited st4te ( 3) to the metastable 5D state ( 4) of the donor Ions and excites acceptors. -
~.
74
K. Tonooka et a!.
/
Journal of Luminescence 62 (1994) 69—76
model (solid curves) give better fits than those by the previous model (dashed curves), as shown in
Table 2 3+ —Tb3 + energy transfer Curve fit parameters of Tb Tb 0
k
k
R
“~
6m4] [mol%] 0.3 1.0
[10’s’] 1.0 1.0
[10’s’] 4 5
[nm] [lO’ 1.12 2.0 1.12 2.0
0.6 0.6
0.0 0.0
4.0
1.0
25
1.12 2.0 1.10 210
0.6 0.4
olo
Fig. 5. With more than 4.0 mol.% of Tb did not draw theoretical curves by the 2previous 03, we model because they were too close to the Y-axis to be clearly seen. It can be concluded that the analysis without taking account of back-transfer under-
The excitation energy of acceptors is dissipated by multi-phonon process [18].5DIn addition to the 5D cross-relaxation between the 3 4 and the 7F,~ transitions, phonon-assisted cross—* 7F 7F 7F 7F relaxation to the 6 1 and the 6 2 transitions were reported 7F 7F [19]. 7FTaking these transitions to the 0, 3 and 2 levels into consideration, the obtained value of the relaxation rate of acceptors, kA ~ 10~at their medium concentration is supported by the report on the multiphonon relaxation in glasses [20]. The estimated dependence of kA on Tb2 03 content can be also understood from the multi-phonon relaxation,
estimates the energy relaxation rate of acceptors. Similar effect of back-transfer was reported for Pr3~ions in a cubic lattice by Huber et al. [14] based on the FOrster theory. The present model gives better fits, although the fits do not always coincide with the experimental data at higher content of Tb 2O3. These errors may be mainly caused by the migration of energy, which were neglected in this analysis. The initial spatial distribution of excitation will change to a spatial equilibrium migration within system. At through high concentration of Tbthe donor 2 03, the average donor separation is small and the probability for energy migration between donors is large. Thus the migration leading to a spatial equilibrium may be important at the initial portion of the fluorescence decay in highly doped glasses. The temperature dependence of the back-transfer was not examined in this study. It will be interesting to investigate the parameter ~ as a function of
4. Effect of back-transfer
—÷
-.+
-÷
With the increase of Tb203, multi-phonon relax-
temperature, since it is suggested that the micro-
ation will be enhanced. Consequently, kA will be greatly increased due to 7F 7F the low energy gap between the 0 and the 6 levels. This interpretation corresponds to the concentration dependence 5D of intensity and lifetime of 4 fluorescence [21].
scopic rates for backward and forward transfer are in the ratio of exp( i~E/kT),where AE is the energy mismatch.
The population density of donors at
5. Conclusions
t
=
0,
Pd(O)
depends slightly on Tb203 content in Table 2. This reduction of pd(O) is understood as a result of the increased absorption of incident light at high concentration of dopants. Thein observed shift[21] of absorption edge of glasses excitationred range is consistent with the concentration dependence of Pd(O). The number of Tb3 ions at the second nearest neighbor position is evaluated to be about 4itflR~C~= 3 at its highest concentration. In order to examine the effect of back-transfer, theoretical expectations by the present model were compared with those by a previous model without back-transfer. The theoretical curves by the present +
—
We propose to improve the previous non-linear model by including back-transfer process de3 + for -doped scribing The the non-linear donor fluorescence glasses. product ofoftheTbdonor and acceptor populations, both forward and backward transfer of energy, and the effect of segregation of donors and acceptors were shown to be the dominant factors characterizing energy transfer in a donor—acceptor system. The transient response was analyzed using the non-linear model including back-transfer, where the efficiency of back-transfer ~ was estimated to be 0.2 from the spectral overlap
K. Tonooka et al.
/
Journal of Luminescence 62 (1994) 69—76
75
of donors and acceptors. A good agreement was observed between the experimental results and theoretical expectations with reasonable energy relaxation rate of acceptors. From the analysis of the 5D 5D 3 4-doped 3— 4thecross-relaxation for for the Tb glasses, critical distance the electric dipole—dipole interaction was obtained as R 0 = 1.1 nm. We plan further investigations to examine the effect of migration of energy, the spatial distribution of excitation density of donors and acceptors, and the temperature dependence of the back-transfer effect.
to the FOrster treatment, the distribution of acceptors should be proportional to the square of r for all r to give the non-exponential decay curves. This non-zero behavior of the distribution function at r ~ 0 results in the appe~ranceof the infinite transfer rate of energy at t = b. It is possible to remedy the infinite transfer rate by introducing a cut-off to
Acknowledgement The authors are indebted to the staffs of research and development division of of Sumita Optical Mfg. Co., Ltd. for the preparation glasses.
the population of acceptors is low enough to be neglected. If the modified distribution with the cutoff is applied to Eq. (A.1), the second of 6 p(t), becomesterm finite. Eq. (A.!), kR~we (RO/Rk) Consequently, get single-exponential decay curves as exact solutions, since it is an ordinary linear rate equation of first-order. And yet, the
Appendix: Differences from Förster-like models
Förster model seems to be a good approximation to the donor fluorescence, since it gives good fits to
In this appendix, we describe the points of the non-linear model compared with the FOrster theory. It is possible to interpret the non-exponential decay of the donor fluorescence as a result of the non-linear dynamics of donor—acceptor interactions. A coupled non-linear rate equation is introduced in the non-linear model, taking account of not only the donor population but also the acceptor population. It should be emphasized that the non-linear model and the FOrster-like models are based on the same pair-transfer function, although, there are great differences in their extensions to include the many-body effect of an actual donor—acceptor system. FOrster started the analysis from the following linear rate equation for the population of donors,
experimental results. There are other questions to FOrster-like models that they are confined to the analysis of fluorescence decays without excitation and that the energy relaxation rate of acceptors is hardly mentioned in these. It is a significant advantage of the non-linear model that it gives solutions both for the transient
dp(t)
— —
kRp(t)
—
kR
~N ( —p) 6 p(t), ‘~
k~i
(A.1)
the acceptor distribution at r ~ 0. It will be even realistic to assume that~there exist no acceptors within the minimum ~lonor—acceptor distance. However, the cutting-off of the acceptor distribution brings a theoretical inconsistency to FOrsterlike models. We accept Eq. (A. 1) as the basic equation for donor—acceptor systems, in the case that
~=
response and for the steady-state of donor fluorescence. Extra parameters such as the energy relaxation rate of acceptors, are introduced in the non-linear model, however, they are explicitly treated with physical picture. The non-linear effect associated with the excitations of donors and acceptors is one of the possible origins causing nonexponential decays of dc~norfluorescence. Theoretical studies on the origins of non-exponential fluorescence are under way. The stochastic approach reported in Ref. [13] will help us to understand the dynamics of donor—acceptor systems.
\RkJ
where Rk is the distance between the donor and the kth acceptor. Eq. (A.1) is restricted to the electric dipole—dipole interaction for simplicity. It should be noticed that Eq. (A. 1) is also obtained as a linear approximation of the non-linear model. According
References [1] Th. Förster, Z. Naturlorsch. 4a (1949) 321. [2] D.L. Dexter, J. Chem. Phys. 21(1953) 836.
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[3] M. Inokuti and F. Hirayama, J. Chem. Phys. 43 (1965) 1978. [4] M. Yokota and 0. Tanimoto, J. Phys. Soc. Jap. 22 (1967) 779. [5] MJ. Weber, Phy. Rev. B 4 (1971) 2932. [6] W.R. Heller and A. Marcus, Phy. Rev. 84 (1951) 809. [7] L.J. Dowell, Los Alamos National Laboratory report LA11873-MS (1990). [8] D.K. Sardar and R.C. Powell, J. Lumin. 22 (1981) 349. [9] H. Shiebold and J. Heber, J. Lumin. 22 (1981) 297. [10] U. Köbler, Z. Phys. 250 (1972) 217. [11] H. Dornauf and J. Heber, J. Lumin. 22 (1980) 1. [12] K. Tonooka, N. Kamata, K. Yamada, K. Matsumoto and F. Maruyama, J. Lumin 50 (1991) 139. [13] K. Tonooka, K. Yamada, N. Kamata, and F. Maruyama, J. Lumin., to be published,
(1994) 69—76
[14] D.L. Huber, D.S. Hamilton and B. Barnett, Phys. Rev. B 16, (1977) 4642. [15] A. Blumen, J. Chem. Phys. 74 (1980) 2632. [16] By. Shulgin, K.N.R. Taylor, A. Hoaksey and R.P. Hunt, J. Phys. C 5 (1972) 1716. [17] A. Hoaksey, J. Wood and K.N.R. Taylor, J. Lumin. 17 (1978) 385. [18] T. Miyakawa and D.L. Dexter, Phys. Rev. B 1 (1970) 2961. [19] N. Bodenschatz, R. Wannemacher, J. Heber and D. Mateika, J. Lumin. 47 (1991) 159. [20] C.B. Layne, W.H. Lowdermilk and M.J. Weber, Phys. Rev. B 16 (1977) 10. [21] K. Tonooka, N. Kamata, K. Yamada, K. Matsumoto and F. Maruyama, Proc. Int. Conf. on Science and Technology of New Glasses (Tokyo, Japan, October 1991) p. 400.
ELSEVIER
JOURNAL OF
LUMINESCENCE Journal of Luminescence 62 (1994) 69—76
Effect of back-transfer on the energy transfer in Tb3 + -doped glasses Kazuhiko Tonookaa, ~, Fumio Maruyama” 1, Norihiko Kamata”, Koji Yamada”, Jun Ono” Hokkaido National Industrial Research Institute, Sapporo 062, Japan b
Department of Functional Materials Science and Engineering, Saitama University, Urawa 338, Japan Received 8 December 1992; revised 18 April 1993, 8 February 1994; accepted 21 February 1994
Abstract The transient response of donor fluorescence under the influence of energy transfer has been analyzed using a non-linear model including the back-transfer effect ofenergy from acceptors to donors. In addition to the non-linear product of donor and acceptor populations and their improved distribution functions, the back-transfer process turned out to be important in describing the interacting donor—acceptor system. It was revealed th~tthe back-transfer ofenergy enhances the non-exponentiality of decay curves. The numerical analysis of energy transfer in Tb3 + -doped glasses showed a good agreement between experimental results and the model, which enables us to discuss the more detailed mechanism of energy transfer.
1. Introduction Energy transfer of electronic excitation from one ion (energy donor) to another (energy acceptor) has been attracting attention both for theoretical interest and for practical use such as phosphors and lasers. Since the fundamental works by Förster [1] and Dexter [2], various models for the energy transfer taking into account multi-polar interaction [2,3], exchange interaction [2,3], diffusion of energy [4,5] and the migration of energy [5,6], have been developed. A detailed bibliography on this field was prepared at the Los Alamos National Laboratory [7].
1
On leave from; Sumita Optical Mfg. Co., Ltd.
*
Corresponding author,
Förster analyzed decay curves of donor fluorescence by a linear rate equation of the donor population, disregarding whet~ieracceptors were excited or not. However, the population of acceptors is expected to play an important role to the energy transfer process with the increase of acceptor concentration. There are several possible models, such as the Inokuti—Hirayam~model [3], the cascade model [8], the discrete shell model [9] and the sum-of-exponentials model [10], for the non-exponential decay curves of donor fluorescence in particular conditions, although none of them give a general solution for au cases. There still remain theoretical problems and discrepancies between models and experimental results. The main difficulties of the Förster and Inokuti—Hirayama models~are the infinite energy transfer rate, which appears at the initial point of
0022-2313/94/$07.0O © 1994 — Elsevier Science By. All rights reserved SSDI 0022-2313(94)00019-9
70
K. Tonooka et a!.
/ Journal of Luminescence 62
decay curves, and the completely random distribution of acceptors, even within the ionic radius, around a donor. The modification of acceptor distribution was already suggested as an approach to form a realistic distribution of active ions by Dornauf et al. [11]. In a previous paper [12], we proposed a non-linear model in which acceptors were assumed to be distributed outside the nearest neighbor distance of the donor and the macroscopic transfer rate of energy was finite at all time. Recently, we have found that the non-linear effect is one of the reasonable origins causing nonexponential responses under the influence energy transfer [13]. The differences between the non-linear model and the Förster model are outlined in the appendix. Numerical calculations showed that the non-linear model gives non-exponential decay curves similar to the Inokuti—Hirayama equation and a good fit to the experimental results of donor fluorescence, although the estimated energy relaxation rate of acceptors was much lower than expected. While there have been many reports on energy transfer between various rare-earth ions, only a few reports dealt with the backward transfer of electronic excitation from acceptors to donors (backtransfer) [14,15]. From the basic formula by Forster and Dexter, the energy transfer rate strongly depends on the overlap of emission and absorption spectra. Since the small Stokes shift of rare-earth ions and the inhomogeneous spectral broadening characteristic to glasses increase the magnitude of the overlap integral, the back-transfer, in addition to the forward-transfer, should be considered in the analysis of energy transfer. In this study, we extended the non-linear model by including backtransfer of energy and analyzed the fluorescence of Tb3 + ions in glasses. A better agreement between experiment and theory was obtained and the value of energy relaxation rate of acceptors became reasonable when compared with the multi-phonon relaxation rate.
2. Formulation We start from the following rate equation [12] for the population of donors and acceptors in the
(1994) 69—76
excited state, pd(t) and pa(t), respectively, under the influence of energy transfer,
dpd(t) dt
=
(1 —
—
Td’.
dPa(t)
—
(kR ~
—
k ‘1
“Tak’
—
—
kN)pd(t)
+
1
t
Pa’. 1/Pd Pa~.tJ)Pd~t) \ — “APa t
,
at where g is the excitation rate of donors, kR the radiative transition rate of donors, kN the nonradiative transition rate of donors, kA the energy relaxation rate of acceptors, kTd the energy transfer rate from a donor to acceptors, kTa the energy transfer rate from donors to an acceptor. In this paper, we extend the above rate equation by including the back-transfer of energy. One of the main factors affecting transfer rates is the overlap integral of emission and absorption spectra as derived by Förster and Dexter. Since the Stokes shift of rate-earth ions is small, the increased overlap integral will lead to an enhancement of the backtransfer. Furthermore, the phonon side band at room temperature [9] and the spectral broadening characteristic of glasses increase the transfer rate of both forward and backward directions. As a consequence, not only the forward-transfer but also the back-transfer play an important role in the energy transfer process. The Stokes shift of Tb3 + is considered to be same as the half width of the emission line shape from experimental results [16,17], as shown schematically in Fig. 1. They are typically about 10 nm in a silicate glass, therefore we can estimate the efficiency of the back-transfer as ~ = 0.2, assuming that the normalized emission spectrum of donors and the normalized absorption spectrum of acceptors are equal and that these line shapes of emission and absorption are Gaussian, as shown in the figure. The back-transfer rate of energy will be proportional to the acceptor population in excited state and the donor population in ground state. Thus, we obtained the rate equation for the population of donors and acceptors including the effect of back-transfer,
K. Tonooka et a!. / Journal of Luminescence 62 (1994) 69—76
0.12
71
_______________
F~=Absorptionof donor ~d=Emission of donQr
0
0.1 0
~d=
—
F
F 5=Absorption of acceptor ta ~Emission of acc8ptor •~ =Overlapof ~°~P°~ ~a8fldFd ~dafldFa
I’
°
a, I
0.08-
Fd/
CI)
~
~
/
0.06-
o
__________
I
0.04
/
-
o
/ I
cr0.02-
I S Fa(?~)dAI S fa(~FdQ’)d)~~O.O1~1 td(
/
z
_________________ +0.0681
/
oCl)
/ / /
0.00 350
360
S..
370
380
390
400
41
420
WAVELENGTH (nm) Fig. 1. Schematic spectra ofabsorptionand emission for donors and acceptors. The forward and back4ransfer rates are proportional to ~J~(a) Faa) d)~and Jf~(2)Fd(~.)dA, respectively. The back-transfer efficiency ‘la = 0.2 is estimated from these overlap integrals, by using a typical data of Tb’ + in glasses [16,17].
dpd(t) dt
=
(1 —
—
pd(t))g
kTd(l
—
—
Radiative Relaxation
(kR + kN)pd(t)
~kR
Pa(t))Pd(t) Forward-Transfer
+ flBkTdPa(t)(l
dpa(t)
dt
—
kTa(1
—
—
Pd(t)),
(2)
g
Pa(t))Pd(t)
—
—
?lBkTaPa(t)(l
—
Excitation
kid
Dono~
I
1B~d~
kia
Acceptors
BkTa
Back-Transfer
Pd(t)) — kAPa(t).
Radiativ kA~
e and Non-radiative
The macroscopic flow of energy is illustrated in Fig. 2. The intensity of the donor fluorescence is proportional to the product of pd(t) and the donor concentration Cd, and is expressed as 1(t)
it
Cdpd(t),
(3)
where it is a coefficient. It is possible to analyze the donor fluorescence 5D 5D using Eqs. (2) and (3). the itcase of 3 -+to set 4 3 +Inions, is reasonable cross-relaxation of Tb donors and acceptors are kTd = kTa = kT because Tb3 + ions. Since the 5D 4 state is the metastable
Non-radiative Relaxation
Relaxation
Fig. 2. A macroscopic model Øf excitation, relaxation, forwardtransfer and back-transfer of ~nergy. A donor gives energy to nearby acceptors and an acceJ~torobtains energy from nearby donors. Since the spatial distri~,utionsof donors and acceptors are generally different, these tr4nsfer rates kTd and k 1~should be distinguished in the analysis.
state of inthethedonor, the I~’ig. ~D43 state be considered analysis. showsshould the schematic flow of energy for the Tb~~—Tb3+ cross-relaxation. In the forward-transfer process, the donor relaxes
K. Tonooka et a!. / Journal of Luminescence 62 (1994) 69—76
72
where k4 is the energy relaxation rate of the 5D4 state. In order to analyze the microscopic proper-
Radiative Relaxation
5D I ExcitatIon
g
at
~+
3 state
d~
P
ties of energy transfer, we use following relations [12],
donors)
~(excited
k1
p Non-radiatIve Relaxation
=
ItDkR
~
+ 4irDk~fl~-~:
(5)
1~4
krII~11eki
and
~ij3
Tb3~at7F
Forward-Transfer ““~
at 5D 4 state
0state
D
=
4c4 3h 44
_____
2C )4
$
t~7a(E)
dE
ç fd(E)1’a(E)
dE,
(K !/
(excited acceptors)
(meta-stable donors)
(6) 3 ions and where R~ is the Cdminimum is the concentration distance between of Tb Tb3 ions, /3 is
,j,kA
+
~1(4 Radiative and Non-radiative
Radiative Relaxation and Non-radiative
+
Relaxation
+
Fig. 3. Schematic flow of ener~ifor the Tb’~—Tb’~cross-relaxation considering the metastable state. In the forward-transfer process, the donor relaxes from its excited state to its metastable state, giving a part ofexcitation energy to a ground state acceptor. In the back-transfer process, the energy of the acceptor transfers back to the metastable donor and thereby excites the donor to the excited state.
from its excited state to its meastable state, giving a part of excitation energy to a ground state acceptor. As a result of the cross-relaxation, the metastable donor and the excited acceptor appear enough in the distance to interact. In the backtransfer process, the energy of the acceptor transfers back to the metastable donor and thereby excites the donor to the excited state. Then 5Dthe rate equation including the population of 4 state p4(t) is given by dpd(t) dt
=
(1 —
pd(t))g
—
k1(1
—
—
(kR +
kN)pd(t)
Pa(t))Pd(t)
+ flBkTPa(t)P4(t),
dPa(t)
dt
=
4(t) dt
kT(l
—
(4)
Pa(t))Pd(t)
1CTPa(t)P4(t) — kAPa(t),
dp
—
=
1B
kT(1 —
a parameter characterizing the segregation of Tb3 ions, n is the refractive index, c is the velocity of light, K is the dielectric constant, e~is the permittiv-
—
Pa(t))Pd(t)
~iB kTP4(t)Pa(t)
—
k4p4(t),
ity of
medium, E is photon energy, dE) is the energy integral of the optical cross-section 7a(E) in the absorption band, fd(E) is the normalized emission spectrum of the donor and Fa(E) is the normalized absorption specQa( =
the
S 7a(E)
trum of the acceptor. We can set the quantity ~/, i/2e 1, equal to unity in the numerical evaluation. The interaction between donors and acceptors is assumed to be electric dipole—dipole in the following analyses for simplicity. In this case, the critical transfer distance R0 is equal to D 1/6
3. Experimental The glass samples under investigation contained Tb2O3 and Na20 in addition to the network formers (NWF) such as Si02, Ge02, Be2 03 and P2 0~.Their compositions are expressed in terms of molar percentages x, y and z as x NWF + y Na2O + zTb2O3, where x + y + z = 100 and y/x = 0.5. The glass samples with z = 0.3 to 7.0 for each NWF were prepared [12] as listed in Table 1. Before measuring the fluorescence intensity, all of the same glasses surface were cut condition. and polished The time in order dependence to ensure of 34 fluorescence intensity from the 5D 5D Tb 3 excited donor state and from the 4 metastable state in silicate glasses were simultaneously measured by the N2 laser (2 = 337 nm) pulse excitation, with the
K. Tonooka et a!.
/
Journal of Luminescence 62 (1994) 69—76
73
Table 1 Sample data of Tb’k-doped glasses NWF
SiO,
Ge0
Base glass composition [mol.%] Sample composition [mol.%] Doped Tb,O, [mol.%] Melting temperature Annealing ( — 20°C/h) Size
67SiO2 + 33Na,O 67GeO, + 33Na,O x NWF + y Na20 + z Tb,O, (x + y + z = z = 0.3, 0.7, 1.0, 1.5, 2.5, 4.0, 6.0 and 7.0 1300°C 1000°C 400—100°C 450—100°C 5 mm x 5 mm x 8 mm
2
1-
-.-
:~ I—
-~ >-
P205
67B2O, + 33’Na20 100, y/x = 0.5)
67P20, + 33Na2O
1200°C 450—100°C
900°C 200—100°C
5D 7F Tb’°( 3-+ 5)
Tb20, 0.3 mol% -~
B20,
~\
So’d curves wIth beck-transfer Dashed curvee:without beck-transfer Dofeexperlnrenbe
9
8 6 4
Z
uJ I—
\5..
z
7~.
Z
w
r5
w 0 z w ()
—
N
N
to w
~
cc O 0.1 -J
~.
U~
~
-
z
7F
cc 0 0.1 e W
°‘~
U) w
‘D—s
05
“1w
._J
Ô.0
I 0.5
I 1.0
I 1.5
4
t
~~ 2.0 20
TIME(msec)
\\
‘~°~
~ 4.Omoi% 50
2.5mol% ~
‘,
\
‘.1
~-.-
100
I 150
200
Fig. 4. Fluorescence decay curves of the 5D, —v ‘F 5 and the and glass. the slow of ‘D4 —v ‘F5 transitions of5D, Tb’fluorescence + in a silicate Therise nonexponential 5D decay of the the 4 fluorescence reflect the energy transfer between the ‘D4 (donor) and the ‘F6 —v ‘F0 (acceptor) transitions.
half width of 1 ns, as shown in Fig. 4. The non5D exponential 5D decay ofthe 3 emission and the slow rise of5D 4 emission reflect 5D the population 5D 5D 7F change 7F from 3 to 4 due to the 3 —‘ 4: 6 0 cross-relaxation, 3 + fluorescence in The decay Tb silicate glassescurves under for thethe influence of cross-relax—~
ation were analyzed numerically using the rate equation (4). All measurements were carried out at room temperature. 5D Fig. 5 shows the experimental points for the 3 ~‘F5emission and the fitted curves obtained by the non-linear analysis including back-transfer. Curve fitting parameters are —
TIME
( iisec
Fig. 5. The ‘D 3 —p ‘F5 (donor) decay curves analyzed using the non-linear model. The theor~tical curves with back-transfer gives better fit than those without back-transfer.
listed in Table = 1000 s and 1 from2.theWe 5Dput kR5D k4 = 250 s 3 and 4 decay curves for 0.3 mol.% of Tb2 0 A distance R0 of 0.4 nm was used,3~ which nitinimum distance between ions isintheTb two Tb rate kN was t~eglectedagainst 203. The non-radiative transition kR. The critical transfer distance R 0 is evaluated instead of the parameter D, assun~ingelectric dipole—dipole interaction. 5D The rapid decay of the fluorescence from the 3 state with increasing Tb2 03 content is due to the cross-relaxi~tion, 5Dwhich transfers electrons from the excited st4te ( 3) to the metastable 5D state ( 4) of the donor Ions and excites acceptors. -
~.
74
K. Tonooka et a!.
/
Journal of Luminescence 62 (1994) 69—76
model (solid curves) give better fits than those by the previous model (dashed curves), as shown in
Table 2 3+ —Tb3 + energy transfer Curve fit parameters of Tb Tb 0
k
k
R
“~
6m4] [mol%] 0.3 1.0
[10’s’] 1.0 1.0
[10’s’] 4 5
[nm] [lO’ 1.12 2.0 1.12 2.0
0.6 0.6
0.0 0.0
4.0
1.0
25
1.12 2.0 1.10 210
0.6 0.4
olo
Fig. 5. With more than 4.0 mol.% of Tb did not draw theoretical curves by the 2previous 03, we model because they were too close to the Y-axis to be clearly seen. It can be concluded that the analysis without taking account of back-transfer under-
The excitation energy of acceptors is dissipated by multi-phonon process [18].5DIn addition to the 5D cross-relaxation between the 3 4 and the 7F,~ transitions, phonon-assisted cross—* 7F 7F 7F 7F relaxation to the 6 1 and the 6 2 transitions were reported 7F 7F [19]. 7FTaking these transitions to the 0, 3 and 2 levels into consideration, the obtained value of the relaxation rate of acceptors, kA ~ 10~at their medium concentration is supported by the report on the multiphonon relaxation in glasses [20]. The estimated dependence of kA on Tb2 03 content can be also understood from the multi-phonon relaxation,
estimates the energy relaxation rate of acceptors. Similar effect of back-transfer was reported for Pr3~ions in a cubic lattice by Huber et al. [14] based on the FOrster theory. The present model gives better fits, although the fits do not always coincide with the experimental data at higher content of Tb 2O3. These errors may be mainly caused by the migration of energy, which were neglected in this analysis. The initial spatial distribution of excitation will change to a spatial equilibrium migration within system. At through high concentration of Tbthe donor 2 03, the average donor separation is small and the probability for energy migration between donors is large. Thus the migration leading to a spatial equilibrium may be important at the initial portion of the fluorescence decay in highly doped glasses. The temperature dependence of the back-transfer was not examined in this study. It will be interesting to investigate the parameter ~ as a function of
4. Effect of back-transfer
—÷
-.+
-÷
With the increase of Tb203, multi-phonon relax-
temperature, since it is suggested that the micro-
ation will be enhanced. Consequently, kA will be greatly increased due to 7F 7F the low energy gap between the 0 and the 6 levels. This interpretation corresponds to the concentration dependence 5D of intensity and lifetime of 4 fluorescence [21].
scopic rates for backward and forward transfer are in the ratio of exp( i~E/kT),where AE is the energy mismatch.
The population density of donors at
5. Conclusions
t
=
0,
Pd(O)
depends slightly on Tb203 content in Table 2. This reduction of pd(O) is understood as a result of the increased absorption of incident light at high concentration of dopants. Thein observed shift[21] of absorption edge of glasses excitationred range is consistent with the concentration dependence of Pd(O). The number of Tb3 ions at the second nearest neighbor position is evaluated to be about 4itflR~C~= 3 at its highest concentration. In order to examine the effect of back-transfer, theoretical expectations by the present model were compared with those by a previous model without back-transfer. The theoretical curves by the present +
—
We propose to improve the previous non-linear model by including back-transfer process de3 + for -doped scribing The the non-linear donor fluorescence glasses. product ofoftheTbdonor and acceptor populations, both forward and backward transfer of energy, and the effect of segregation of donors and acceptors were shown to be the dominant factors characterizing energy transfer in a donor—acceptor system. The transient response was analyzed using the non-linear model including back-transfer, where the efficiency of back-transfer ~ was estimated to be 0.2 from the spectral overlap
K. Tonooka et al.
/
Journal of Luminescence 62 (1994) 69—76
75
of donors and acceptors. A good agreement was observed between the experimental results and theoretical expectations with reasonable energy relaxation rate of acceptors. From the analysis of the 5D 5D 3 4-doped 3— 4thecross-relaxation for for the Tb glasses, critical distance the electric dipole—dipole interaction was obtained as R 0 = 1.1 nm. We plan further investigations to examine the effect of migration of energy, the spatial distribution of excitation density of donors and acceptors, and the temperature dependence of the back-transfer effect.
to the FOrster treatment, the distribution of acceptors should be proportional to the square of r for all r to give the non-exponential decay curves. This non-zero behavior of the distribution function at r ~ 0 results in the appe~ranceof the infinite transfer rate of energy at t = b. It is possible to remedy the infinite transfer rate by introducing a cut-off to
Acknowledgement The authors are indebted to the staffs of research and development division of of Sumita Optical Mfg. Co., Ltd. for the preparation glasses.
the population of acceptors is low enough to be neglected. If the modified distribution with the cutoff is applied to Eq. (A.1), the second of 6 p(t), becomesterm finite. Eq. (A.!), kR~we (RO/Rk) Consequently, get single-exponential decay curves as exact solutions, since it is an ordinary linear rate equation of first-order. And yet, the
Appendix: Differences from Förster-like models
Förster model seems to be a good approximation to the donor fluorescence, since it gives good fits to
In this appendix, we describe the points of the non-linear model compared with the FOrster theory. It is possible to interpret the non-exponential decay of the donor fluorescence as a result of the non-linear dynamics of donor—acceptor interactions. A coupled non-linear rate equation is introduced in the non-linear model, taking account of not only the donor population but also the acceptor population. It should be emphasized that the non-linear model and the FOrster-like models are based on the same pair-transfer function, although, there are great differences in their extensions to include the many-body effect of an actual donor—acceptor system. FOrster started the analysis from the following linear rate equation for the population of donors,
experimental results. There are other questions to FOrster-like models that they are confined to the analysis of fluorescence decays without excitation and that the energy relaxation rate of acceptors is hardly mentioned in these. It is a significant advantage of the non-linear model that it gives solutions both for the transient
dp(t)
— —
kRp(t)
—
kR
~N ( —p) 6 p(t), ‘~
k~i
(A.1)
the acceptor distribution at r ~ 0. It will be even realistic to assume that~there exist no acceptors within the minimum ~lonor—acceptor distance. However, the cutting-off of the acceptor distribution brings a theoretical inconsistency to FOrsterlike models. We accept Eq. (A. 1) as the basic equation for donor—acceptor systems, in the case that
~=
response and for the steady-state of donor fluorescence. Extra parameters such as the energy relaxation rate of acceptors, are introduced in the non-linear model, however, they are explicitly treated with physical picture. The non-linear effect associated with the excitations of donors and acceptors is one of the possible origins causing nonexponential decays of dc~norfluorescence. Theoretical studies on the origins of non-exponential fluorescence are under way. The stochastic approach reported in Ref. [13] will help us to understand the dynamics of donor—acceptor systems.
\RkJ
where Rk is the distance between the donor and the kth acceptor. Eq. (A.1) is restricted to the electric dipole—dipole interaction for simplicity. It should be noticed that Eq. (A. 1) is also obtained as a linear approximation of the non-linear model. According
References [1] Th. Förster, Z. Naturlorsch. 4a (1949) 321. [2] D.L. Dexter, J. Chem. Phys. 21(1953) 836.
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K. Tonooka et al.
/ Journal of Luminescence 62
[3] M. Inokuti and F. Hirayama, J. Chem. Phys. 43 (1965) 1978. [4] M. Yokota and 0. Tanimoto, J. Phys. Soc. Jap. 22 (1967) 779. [5] MJ. Weber, Phy. Rev. B 4 (1971) 2932. [6] W.R. Heller and A. Marcus, Phy. Rev. 84 (1951) 809. [7] L.J. Dowell, Los Alamos National Laboratory report LA11873-MS (1990). [8] D.K. Sardar and R.C. Powell, J. Lumin. 22 (1981) 349. [9] H. Shiebold and J. Heber, J. Lumin. 22 (1981) 297. [10] U. Köbler, Z. Phys. 250 (1972) 217. [11] H. Dornauf and J. Heber, J. Lumin. 22 (1980) 1. [12] K. Tonooka, N. Kamata, K. Yamada, K. Matsumoto and F. Maruyama, J. Lumin 50 (1991) 139. [13] K. Tonooka, K. Yamada, N. Kamata, and F. Maruyama, J. Lumin., to be published,
(1994) 69—76
[14] D.L. Huber, D.S. Hamilton and B. Barnett, Phys. Rev. B 16, (1977) 4642. [15] A. Blumen, J. Chem. Phys. 74 (1980) 2632. [16] By. Shulgin, K.N.R. Taylor, A. Hoaksey and R.P. Hunt, J. Phys. C 5 (1972) 1716. [17] A. Hoaksey, J. Wood and K.N.R. Taylor, J. Lumin. 17 (1978) 385. [18] T. Miyakawa and D.L. Dexter, Phys. Rev. B 1 (1970) 2961. [19] N. Bodenschatz, R. Wannemacher, J. Heber and D. Mateika, J. Lumin. 47 (1991) 159. [20] C.B. Layne, W.H. Lowdermilk and M.J. Weber, Phys. Rev. B 16 (1977) 10. [21] K. Tonooka, N. Kamata, K. Yamada, K. Matsumoto and F. Maruyama, Proc. Int. Conf. on Science and Technology of New Glasses (Tokyo, Japan, October 1991) p. 400.