E X C I T E D S T A T E P H E N O M E N A IN V I T R E O U S M A T E R I A L S 41
between different ions or ions of a similar nature. The last transfer is also named cross-relaxation.
The radiative transfer consists of absorption of the emitted light from a donor molecule or ion by the acceptor species. In order that such transfer takes place, the emission of the donor has to coincide with the absorption of the acceptor. The radiative transfer can be increased considerably by designing a proper geometry.
Such transfers may be important in increasing pumping efficiencies of glass lasers (Reisfeld, 1985a) and luminescent solar concentrators, whereby energy emitted from organic molecules can be absorbed by ions such as Cr(III), Mn(II), or Nd(III), followed by characteristic emission from these ions.
42 R. REISFELD and C.K. JORGENSEN
and JOrgensen, 1977) that the glasses and crystalline materials most favorable for luminescence tend to have the highest (fundamental frequency) phonon hto with as low wave number as possible, providing smaller rates W. Though the parame- ter a has the dimension of a reciprocal wave number, it is not simply 1/hw, but is related by a small constant a in a = (aho)) -~, where a is increasing from 0.25 for LaCI 3 through 0.4 for aqua ions and the garnet Y3A15Oa2 up to 0.5 for Y203.
Crystalline LaF 3 has a = 0.5 and Z B L A glass has a -- 0.38. Though eq. (18) has a somewhat greater percentage uncertainty than the Judd-Ofelt treatment of ~'R, it is noted that the parameters B and a do not d e p e n d on the lanthanide chosen (Reisfeld, 1985b,c).
4.1. Multiphonon relaxation
Today, multiphonon relaxation in lanthanide ions is a well-understood process, contrary to other transition metal ions, which still req.uire additional understand- ing. Excited electronic levels of rare earths R in solids decay nonradiatively by exciting lattice vibrations (phonons). When the energy gap between the excited level and the next lower electronic level is larger than the phonon energy, several lattice phonons are emitted in order to bridge the energy gap. It was recognized that the most energetic vibrations are responsible for the nonradiative decay since such a process can conserve energy in the lowest order. In glasses the most energetic vibrations are the stretching vibrations of the glass network polyhedra;
it was shown that these distinct vibrations are active in the multiphonon process, rather than the less energetic vibrations of the bond between the R and its surrounding ligands. It was demonstrated that these less energetic vibrations may participate in cases when the energy gap is not bridged totally by the high-energy vibrations. The experimental results reveal that the logarithm of the multiphonon decay rate decreases linearly with the energy gap, and hence with the number of phonons bridging the gap, when the number of phonons is larger than two. Figure 3 presents the logarithm of the nonradiative relaxation rate versus the number of phonons of the glass-forming material. Figure 4 presents the same dependence;
however, two phonons are taken as the origin of the abscissa.
Application of the multiphonon theory to glasses requires the knowledge of the structural units forming the glass. Similar to the electronic spectra in glasses, the vibrational frequencies show inhomogeneous broadening due to the variation of sites. Table 2 shows the average frequencies of the network formers. The vibrations involving the network modifiers are lower by a factor of 2 to 4.
Considering the dependence of the multiphonon relaxation rate, Wp, on the phonon occupation number for a p-order multiphonon decay at temperature T > 0 ,
Wp = B[n(T) + 11 p e x p ( - ~ A E ) , (19)
EXCITED STATE PHENOMENA IN VITREOUS M A T E R I A L S 43 15
'''*'4.
"-- ~ ~ - ' , ~ . . .... .~.----5
0 5 - t I . . .
O I ~ ~ " ~ ' ~ ' ~
0 0 I 2 5 4 5 6 7
N U M B E R OF P H O N O N S
Fig. 3. Dependence of multiphonon relaxation on the number of phonons in the glass former needed to bridge the energy gap between the luminescent J-level and the closest lower-lying level. The curves refer to glasses of the following types: (1) phosphate; (2) borate; (3) silicate; (4) tellurite; (5) fluoroberyUate; (6) germanate; (7) zirconium fluoride (ZBLA); (8) aluminum lanthanide sulfide (ALS) and gallium lanthanide sulfide (GLS); (9) the crystalline hosts Y3A15012(YAG), and (10)
LaF 3 .
.0'. i o
L
u_ 5 - ~ , ~ , = ~ . , . ~ ... ~.~-~7 o ~ - ~ - ~ . ~ ~ - - : : ~ -' ~ ~... ~ .,.
o('9 ~ 6 ~ ' < ~ ' , ~ : ~ _ _ . 9 ""'~'",,
2 3 4 5 6 7
NUMBER OF PHONONS
Fig. 4. Dependence of multiphonon relaxation on the number of phonons in the glass former and in two crystalline materials (note that the abscissa starts at 2 phonons). The numbering of the samples is
the same as in fig. 3.
44 R. REISFELD and C.K. JORGENSEN TABLE 2
Parameters for nonradiative relaxations in glasses.
',\
Matrix B a ho) a log10 B* Ref. ,3
(s -1) (10 -3 cm) (cm -1)
tellurite 6.3 x 10 l° 4.7 700 0.30 7.97 a
phosphate 5.4 x 1012 4.7 1200 0.18 7.88 a
borate 2.9 x 1012 3.8 1400 0.19 7.89 a
silicate 1,4 X 1 0 az 4.7 1100 0.19 7.89 b
germanate 3.4 x 10 l° 4.9 900 0.23 6.74 a
ZBLA ,1 1.59 x 10 TM 5.19 500 0.38 7.97 c
1,88 x 101° 5.77 460-500 0.36 7.89 d, e
LaF~ (cryst.) 6.6 x 108 5.6 350 0.50 7.1 f
ALS, GLS ,2 106 2.9 350 1.0 5.1 a
, a ZBLA glass: 57 ZrF 4 : 34 BaF 2 : 3 LaF 3 : 4 A1F s : 2 RF 3 .
,z ALS glass: 3 A12S 3:0.872 La2S 3:0.128 NdzS3; GLS glass: 3 Ga2S 3:0.85 LazS 3:0.15 Nd2S 3.
,3 References:
(a) Reisfetd (1980). (d) Shinn et al. (1983).
(b) Layne et al. (1977). (e) Tanimura et al. (1984).
(c) Reisfeld et al. (1985a). (f) Riseberg and Moos (1968).
w h e r e A E is t h e e n e r g y g a p b e t w e e n t h e e m i t t i n g a n d next l o w e r level
a = - l n ( e ) / h w , (20)
a n d a a n d B are d e p e n d e n t o n t h e h o s t b u t i n d e p e n d e n t o f the specific e l e c t r o n i c level o f R f r o m which the d e c a y occurs.
I n calculating t h e o r e t i c a l l y the m u l t i p h o n o n r e l a x a t i o n ( E n g l m a n , 1979), the e l e c t r o n i c m a t r i x e l e m e n t s a r e s e p a r a t e d f r o m t h e vibrational ones, a s s u m i n g t h a t t h e transition is d o m i n a t e d b y a small n u m b e r o f p r o m o t i n g m o d e s w h i c h c o n s u m e t h e e n e r g y hw. T h e a c c e p t i n g m o d e s p r o v i d e only f o r the r e m a i n i n g e n e r g y d i f f e r e n c e A E 0 - h o ) m a x.
T h e m u l t i p h o n o n d e c a y r a t e f r o m a given level to t h e next l o w e r level d e c r e a s e s with the l o w e r i n g o f e n e r g y o f t h e stretching f r e q u e n c i e s o f the glass f o r m e r . Since, in o r d e r to r e a c h t h e s a m e e n e r g y g a p , a l a r g e r n u m b e r o f p h o n o n s is n e e d e d in fluoride a n d in c h a l k o g e n i d e glasses, t h e n o n r a d i a t i v e relaxations are, u n d e r e q u a l c i r c u m s t a n c e s , smallest in these glasses.
T a b l e 2 gives B a n d a o f eq. (19) t o g e t h e r with t h e highest p h o n o n e n e r g y f o r a v a r i e t y o f glasses a n d crystals. F o r m o s t oxide glasses, a is fairly c o n s t a n t while B differs b y several o r d e r s o f m a g n i t u d e . S c h u u r m a n s a n d Van D i j k (1984) carefully a n a l y z e d this p h e n o m e n o n , a n d gave specific r e a s o n s to d e c r e a s e the e n e r g y gap b y twice t h e p h o n o n e n e r g y , a n d h e n c e write
WN R = B* e x p [ ( - A E + 2 h w ) a ] ,
log10 B * = log10 B - 0 . 8 6 a h o ) = loglo B - ( 0 . 8 6 / a ) ,
(21)
E X C I T E D S T A T E P H E N O M E N A IN V I T R E O U S M A T E R I A L S 45
~o
0
o ' ~
0
• = ~<
.+~
i
J
1 ~ 5 I I t ~
o
~5
~ l o °
a !
o o l o o ~ i i i i i
~ V
~_+~,.~ I t*.+ I I ~-~ I I I ~ , - ~ I I I I I I I I I I I ~ - I ~ I I I I I I I I
t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t
+
t ~ '+¢:3
C ~
<.
0
m 3
o
o = o
~ . o
I=I o ~
46 R. REISFELD and C.K. J O R G E N S E N
<~
I,I
i
Q
~ •
~ ' ~
~ Q
o!
" ~
e~
.,o
Q 1
I_
7
!
0
O~ ~4
1 o
i ! '
I I I 0
@4
1 1 e q I
I I I I
i ! i i i i i i ! i ! i i ! i ~ i i i --=. =
i i i l i i = i i i = = i i i = ~ - 7 i i
S
I
N.~t,
% •
EXCITED STATE PHENOMENA IN VITREOUS MATERIALS 47 where 0.86 = lOgl0 e 2. The value of log10 B* is also presented in table 2 and is seen to differ much less between most glasses (and crystals) than B. One may also compare with fig. 4.
The multiphonon transition rates for lanthanides in a variety of glasses can be obtained from eq. (21) using the tabulated B*, a, and hw in table 3 and the energy difference to the next lower J level AE. The good agreement with experiment (typical uncertainties are 0.1 or 0.2 in lOgl0 WNR ) has been demon- strated for Pr(III) (Eyal et al., 1985); Nd(III) (Lucas et al., 1978), Ho(III) (Tanimura et al., 1984; Reisfeld et al., 1985a), and Er(III) (Reisfeld and Eckstein, 1975a; Shinn et al., 1983; Reisfeld et al., 1983a).
The multiphonon processes compete with luminescence as excitation energy is lost to the local vibrations of the glass formers. Other nonradiative losses arise from energy transfer to the electronic states of atoms in the vicinity of the excited ion. The energy-transfer process may be either resonant or phonon-assisted, where the excess of energy is dissipated as heat. The energy deficiency in nonresonant processes may be supplied by the thermal reservoir of the low-energy phonons ( k T = 2 1 0 c m -1 at 300K) to match the missing energy. Because of thermodynamical considerations the last process is much less efficient. As charac- teristic examples of observed and radiative lifetimes, branching ratios, etc., the 4f ~° holmium(III) in Z B L A glass is treated in table 3, and 4f 11 erbium(III) in table 4.
4.2. Cross-relaxations
A special case of energy transfer is cross-relaxation, where the original system loses the energy (E 3 - E2) by obtaining the lower state E 2 (which may also be the ground state EI) and another system acquires the energy by going to a higher state E;. Cross-relaxation may take place between the same lanthanide (being a major mechanism for quenching at higher concentration in a given material) or between two differing elements, which happen to have two pairs of energy levels separated by the same amount.
The cross-relaxation between a pair of R ions is graphically presented in fig. 5.
The two energy gaps may be equal or can be matched by one or two phonons.
Cross-relaxation has been measured in a variety of ions and it is a dominating factor in nonradiative relaxations at high concentration. The nonradiative relaxa- tion rates can be obtained by analysis of the decay curves of R fluorescence using the formula of the general form where the population number of state i, Ni, is proportional to the intensity of emitted light, Ii:
d ~ - Y• + Xi + ~ Wq Ni(t ) + ~, WjiNi(t ) . (22)
i~:j i ~ j
dN/(t)/dt is the decrease of intensity after pulse excitation, YR is the reciprocal of
48
> -
t
(..9 ILl Z LLI
CROSS
R. REISFELD and C.K. J O R G E N S E N
R E L A X A T I O N 3'
2 2 '
Ion A
1 3 > - - - 1 2 > ~ I
Ion A'
> ---I 2'>
Fig. 5. Scheme for cross-relaxation b e t w e e n two i o n s of the s a m e , or of different nature.
the lifetime of the excited state in the absence of a cross-relaxation process. E Wij is the probability for cross-relaxation, W# is the probability of the inversed process, and W~j is the rate of cross-relaxation.
Theoretically, the cross-relaxation rate for a dipole-dipole transfer can be obtained from the formula (Reisfeld, 1976a)
p(DD) = 1 2 (~_~)( eZ ]2 [ ~ t ]
SA (2J s + 1)(2J A + 1) 3 R3/
ats<JsllU(')llJ >2
Here g2 t are the Judd-Ofelt intensity parameters,
<JllU(°llJ'>
is the matrix element of the transition between the ground and excited state of the sensitizer and activator, respectively. [The calculation of these matrix elements in the intermediate-coupling scheme is now a well-known procedure and may be found in Reisfeld (1976a) and Riseberg and Weber (1976)]. S is the overlap integral and R is the interionic distance.The measured lifetime of luminescence is related to the total relaxation rate by
1 ~ W N R + ~ , A + p S R 1 + P c R , (24)
- - = = - -
r %
where E A is the total radiative rate, E WN R is the nonradiative rate and PCR is the rate of cross-relaxation between adjacent ions.
The possible cross-relaxation channels for Pr(III), Nd(III), and Ho(III) are outlined in table 5 (Reisfeld, 1985c, Reisfeld and Eyal, 1985).
In order to obtain the critical radii, which are the distances at which the probability for cross-relaxation is equal to the sum of the radiative and multi- phonon probabilities, the decay curves are analyzed with respect to the Inokuti- Hirayama model (Inokuti and Hirayama, 1965).
EXCITED STATE P H E N O M E N A IN VITREOUS MATERIALS 49 .=.
C~
=
"O
©
eto
O
r~
t~
£}
r.)
+1 +1 +1 +1 +1 +1 ÷1 +1
J r rq eq eq
eq ~
0~
+
, _ . =
N
t~
O
- r .
t'4
*
50 R. REISFELD and C.K. JORGENSEN
The energy transfer between Cr(III) and Nd(III) (Reisfeld and Kisilev, 1985), and Cr(III) and Yb(III) (Reisfeld, 1983), has been studied extensively by static and dynamic methods. The transfers occur by dipole-dipole mechanisms and are important in increasing the pumping range of Nd(III) and Yb(III) ions.
4.3.
Temperature dependence and lifetime of the excited states
Huang and Rhys (1950) base their theoretical explanation of the temperature dependence of the nonradiative decay rate on a single-configuration-coordinate model, in which the parabolas representing the ground- and excited-state oscil- lators both have the same force constants, but whose minima are offset. The use of the time-dependent perturbation theory for calculation of. multiphonon emis- sion rates for the rare-earth ions, with no restriction on force constants or offsets, was also provided (Struck and Fonger, 1975; Fonger and Struck, 1978). In this treatment the theoretical data may differ by one order of magnitude from the experimental results. A phenomenological model of multiphonon emission in crystals (Riseberg and Moos, 1968) and in glasses (Reisfeld and Eckstein, 1975a) gives good agreement with experiment when the number of phonons bridging the emitting and terminal levels is more than two. Recently, a modified energy-gap model was developed (Schuurmans and Van Dijk, 1984). Their model assumes that the phonons having single highest frequencies are active in the nonradiative transition.
The expression for the temperature dependence is
[ exp(hw)/kT ]P
WN (T) = WN (0) L 1 ' (25)
where p is the number of phonons emitted in the transition, and WNR(0 ) is the low-temperature multiphonon emission rate.
Experimentally, the multiphonon emission rate of an excited state i to an adjacent lower-lying state j in the absence of energy transfer is given by
1 1
WNR -- - - (26)
"Fobs ~-i
at a constant temperature. In this formula ~oUs is the observed lifetime of state i at the particular temperature, and the subscript N R stands for the nonradiative process.
Expression (26) has been found to account fairly well for the temperature dependence of the nonradiative rate as determined by the use of equation (25) at various temperatures in oxide glasses (Layne et al., 1977; Reisfeld and Eckstein, 1975a; Reisfeld and Hormadaly, 1976) in cases where the highest-energy phonons were active. However, when the energy gap cannot be bridged by only one
EXCITED STATE PHENOMENA IN VITREOUS MATERIALS 51
frequency phonon, two vibrational modes are responsible for the relaxation (Hormadaly and Reisfeld, 1977; Reisfeld et al., 1978). In instances where less than one phonon is needed to bridge the energy mismatch, the relaxation is slowed down due to a bottleneck created, as shown in the relaxation of Ho(III) in tellurite glass (Reisfeld and Kalisky, 1980a). The temperature dependence of multiphonon relaxation in fluoride glasses was studied for Er(III) (Shinn et al., 1983), for Ho(III) (Tanimura et al., 1984), and for Nd(III) (Lucas et al., 1978).
The temperature dependence in these glasses exhibits a similar behavior to the oxide glasses; however, the phonon energies of about 500 cm -1, characteristic for fluoride glasses when put into eq. (25), result in the best theoretical fit.
5. Energy transfer between differing species