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The non-radiative energy transfer in high acceptor concentration codoped Nd,Ho:YAG and Nd,Er:YAG
Abstract
For t?xw luminese~nt laser ~~s~~s it is fmmd that the d~~~~-dj~o~e jntem~~~~nbetween d~n~r-a~~e~t~r pairs at short distances is damina~d by hewer order multipa~ar ~ntemetions. This effect is refiected as a fast initial decay in the donor emission ~sient of the codoped systems Nd,Ha:YAG and Nd,ErzYAG at high acceptor ~~n~en~ti~ns. The effect is enhanced as the acce@or c~cen~~~n is ‘ii~~~sed, At iong times the donor emission transients show tile effects of the dip&e-dipole interaction between ~~~~~-acce~~~rpairs, that is, the long range multipolar interaction drives the non-radiative energy transfer, The Monte Carlo model shows these effects for the non-radiative energy transfer process applied to those luminescent systems. For this modei the discreteness of the YAG lattice is kept since dopants are randomly placed into the 24c sites. The donor emission ~ns~~~~ are obtained by calculating the ~~~a~~i~t~es of the events that accur after a donor is excited.
The RaR~radiati~eeReTgy transfer pracess has played a major rofe in jnc~~~n~ t&e ~urnpj~~ efficiency of active ions in. soEd state hX%rmatetials ffb An ~~te~~~t~o~between two ions, one in its ground state (the zceptor A) and the other in some excited state (the donor D), drives this process. As a result of the process the excitation moves from ions D to A. Two specific non-radiative isotropic D-A interactions have been considered, namely, rnu~tip~l~ and exchange. The first is an ~nt~rac~~n d~c~m~sabIe into dj~Ie-d~~~~ Id-d), ~~~e-qu~dmpol~ (d-q), qua~p~~e-quadm~~~
The mode1 developed by Forster and Dexter [2J] CFDI for the nob-r~iat~~e energy transfer considers t&at dopants are ~~d~rn~~ placed in the frost material ~eve~~~ess~ f@r tfie ~a~~jca~ expression of this model a cantinuous dist~bution of A ions is assumed, i-e,, for the case of laser crystal rnat~~~a~s the discreteness of tke Mice is neglected. For I& model it is not possible to consider a mixture of interactions between the interacting B-A pairs. FLITthermore, this madei was developed for fow dopant concen~atjons, where D-A average distances ;fpe? large. Despite these drawbacks the FD mode1 has
0030-40~1/96/$IZ,W copyright 0 I %Xi EIsevier Science I3.V”Aft rights reserved. Pff s0030-40 18(96)00062-4
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found wide use for the analysis of experimental data in laser crystals since it gives the basic physics of the transfer process. With a fast detection system Stalder and Bass [4] were able to measure a fast initial decay of the 4F,,, state of neodymium ions in three luminescent systems: Nd(l%),Ho(lO%):YAG, Nd(l%), Er(lS%):YAG and Nd(l%),Er(30%):YAG. For these systems the D and A ions are Nd+3 and Erf3 or Ho+~, respectively. As expected, the FD model did not give a complete understanding of the kind of interaction that was driving the transfer. They turned to the model developed by Golubov-KonobeevSakun (GKS) [5,6] for the non-radiative energy transfer. It was reported that the neodymium fast initial decay was due to a short-range interaction involving the nearest-neighbor sites. As a result they assumed an exchange interaction for short D-A distances and for further distances a d-d interaction. The fraction of D ions that decayed during the fast initial decay was determined from the experiment and with the GKS model. Thus, they concluded that A ions placed in the four nearest-neighbor sites were responsible for the fast initial decay. It is known that d-d interactions can generally be expected to dominate in systems ch~acteri~d by allowed electric dipole radiative transitions [3]. Axe and Weller [7] have pointed out that, although the radiative transitions within the 4f” rare earth configurations are generally electric dipole in nature they are much weaker than fuIly allowed dipole transitions. As a result quad~pol~ processes become of greater importance and can completely dominate non-radiative transfer processes in certain systems. This is more probable to occur with luminescent systems at high A concentrations where the D-A separation is small [S]. The experimental Nd*3 decay curves measured by Stalder and Bass seem to be a clear example where high order multipolar interactions between D-A pairs dominate the d-d interaction at short distances. We support this prediction with Monte Carlo (MC) simulations. In this work those experimental neodymium decay curves are analyzed with the MC model for the non-radiative energy transfer process [9]. For the luminescent systems, Nd(l%),Er(15%3):YAG and Nd(l%),Er(30%):YAG, it is found that the energy transfer is driven by two multipolar interactions, a
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q(R)-q(R) and a d(R)-d(R). For the case of Nd( 1%),Ho( IO%):YAG the driving interactions are a d(R)-q(R) and a d(R)-d(R). For each luminescent system the interactions compete to each other, and the stronger and shorter range interaction dominates the d(R)-d(R) interaction for short D-A distances. This effect is reflected as the fast initial Nd+3 decay measured experimentally, that is expected in the MC simulations. At large D-A distances the d-d interaction dominates the transfer process and the D emission transients calculated by the MC model also follow the experimental decay trends at long times. These effects are enhanced as the A concentration increases.
2. The Monte Carlo (MC) model The MC model for the non-radiative energy transfer process in laser crystals has been reported elsewhere [9], and in here we give a brief account of it. The crystal structure of YAG is 1a3d (230) and the site positions of their components are well known [IO]. When this crystal is doped with rare earth elements they substitute the yttrium ions at 24~ sites. This model considers a cube containing 1000 unit cells of YAG as a working unit that is replicated to fill all space. By using the MC method dopants are randomly placed in the potential crystal sites of this cubic crystal sample. Afterwards they are randomly labeled as D and A ions. With this random placement each D ion has a particular dis~bution of A ions. An interaction between D-A pairs is assumed and the probabilities of the events that occur after a D ion is excited are calculated, i.e., the D and A emission transients. For this work we run 192000 excitations for each codoped crystal. That is, the MC model results, for each sample crystal, are the averages of 800 codoped crystals and exciting each D ion at t=O. For a Coulomb multipole interaction between D-A pairs the transfer rate between the &h-D ion and the jth-A ion, separated by Rlj, is defined by
WDA(Rij)
=
(1)
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Distribution
I
I
I
I
’
I
I
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probability of nearest neighbors YAG: D*‘, AC3 I
/
‘I
/
/,‘,I/,‘//,,
0.8
x
R31%
l
A”
10% A'" 15%
l
A*‘30%
l
X
l
II
x
xx
X
x
X=Rda/Rmin Fig. 1. Probability distribution of nearest neighbors at four A concentrations in codoped D+‘,A+‘:YAG. (D+$ and A+’ stand for trivalent donor and acceptor ions for Nd+3 and Hof3 or Er+3, respectively.) Dopants were randomly placed into 24~ sites of the YAG lattice and afterwards were randomly labeled as D and A ions. The D ions were fixed at 1% concentration and A ions at 1, 10, 15 and 30%. The vertical axis is the probability PD,,, (X,) of finding the nearest D-A distance between X, and X, + 0.2, where X, is adimensional and defined as the ratio between the nearest D-A pair distance RDA,n and the minimum distance between the 24~ sites. The scatter of the plot is due entirely to the discreteness of the lattice.
where Cfsf are the microscopic interaction parameters and the values s = 6, 8, 10,. . . correspond to a d-d, d-q, q-q,. . . interactions, respectively. The microscopic interaction parameter for a particular interaction is defined as Cfs) = (ROS)S/rnO, where R,, is the critical transfer distance defined as the separation between D-A pairs for which the energy transfer rate is equal to (r&Y’, and in,, is the D
life time when A ions are not present. Note that the MC model can consider a single multipolar interaction, as the FD model does, or a mixture of interactions to drive the energy transfer. Once dopants are randomly placed, one can calculate any of the terms in IQ. (1) or assume any mixture between them. To calculate any of the terms in Eq. (1) one has to look for the best R,, value to fit the experimental decay
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Table 1 C”’ values used in the MC simulations for the fluorescence decay of the 4F,,, state of neodymium ions in Nd(l%),Ho(lO%):YAG, Nd(1 %),Erl30%):YAG and Nd(l %),E1(30%):YAG. The C(I) values are defined by (R,)“/T,~, where R, = 9.12 and 8.65 i for the systems with holmium and erbium, respectively, and rot = 240 t.~s. Note that the R, value for holmium is constant for the competing interactions d-q and q-q. For erbium the R, value is constant for the competing interactions d-d and q-q and for both concentrations. The units for C’“’ values am in cm’/s
~(6) c(8) c(lO)
HdlO%)
Ert15%)
Ert30%)
2.39x 10-39 1.99x 10-53 _
1.75x 10-39 -
1.75x
0.98x
0.98X 1O-67
1O-67
10-39
curve. Clearly if a mixture of interactions is needed a single R,, value should be found for all interactions. The total emission transient will be the individual emission transient, averaged over the D ions which are excited at t = 0, namely 4,(t)
= (exp[-t/(T,ff)i])Av.overi,
(2)
where (rerrji is the effective lifetime for the ith
1
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129 (1996) 273-283
donor, defined by 1 (‘eff)i
1 = -
+ 2 W,,( R,J
'DO
j=l
(3)
where the sum is over the number of A ions, NA,and s = 3, 4, 5,. . . for the multipolar interaction between D-A pairs. The average of Eq. (2), i.e., the averaging of the responses of many excited donors each having its own (T~~~)~, is done through probability expressions of the corresponding emissions. As shown in Ref. [9] the average of Eq. (2) is non-exponential. This behavior is a direct result of the average procedure and of the assumed interactions between D-A pairs. If the interaction between D-A pairs is neglected, the second term in Eq. (3) is zero and Eq. (2) gives an exponential behavior, as expected [9].
I-
0.1
Zl 0’ 3 Z Z 5 0.01
0.001
0
40
80 Time
120
(psec)
Fig. 2. Experimental curve and MC-multipolar simulations of the fluorescence decay of the 4Fs,2 state of neodymium ions in Nd(l%),HdlO%):YAG at long and short times. Horizontal axis is time in psec. The vertical axis is intensity in au. showing three orders of decays at long times, and the inset is in natural scale. The solid curve corresponds to a MC simulation for a d-d interaction between D-A pairs. The MC dashed curve is for the case when a d-q and a d-d interaction compete to each other according to the energy transfer rate constant calculated with Eq. (4). The transfer rate for the d-q interaction was calculated up to the first D-A neighbor pairs.
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21-l
the concentration of A ions. Conversely, as the A concentration decreases, e.g., A ions at l%, there is an increasing probability that no A ions exist at small values of X, and that the next D-A pair is at some large X, value. Thus, in the case of A ions at 15%, Fig. 1 shows that almost 50% of D ions have at least an A ion as first neighbor, nearly 40% have an A ion as second neighbor, and so on. For a particular multipolar interaction between D-A pairs, the energy transfer rate was calculated according to NAP(S)
3. Results
Once the YAG crystal is codoped, each D ion has a particular distribution of A ions, and the distribution probability of nearest D-A pairs is calculated. In other words, after codoping the crystal each D ion is inspected for the distance to its nearest A ion, R DA.n. This distante is made non-dimensional by dividing it by 3.67 A - the shortezt distance between 24c sites; thus, X, = R,,,,/3.67 A. The RI)*,” value is counted in its own bin between X, and X, + 0.2, with X, incrementally increasing from 0.0 to 9.8. Fig. 1 shows this probability distribution for nearest neighbors, PDA,n(Xn), at four A ion concentrations, l%, lo%, 15% and 30%. The D ion concentration is fixed at 1%. The functions PDA,n(Xn) shown in Fig. 1 are not continuous because of the discreteness of the YAG lattice. The two zero probabilities at X, = 1.2 and 1.4 correspond to distances which are not present in the lattice. Also, the figure shows an increment in the probability of nearest D-A neighbor pairs with
(4) where the sum is over NA, the total number of A ions. The critical transfer distance, R,, = R,, is the only free parameter for the MC model and it was constant and unique, 9.12 and 8.65 A for Nd,Ho:YAG and Nd, Er:YAG, respectively. The R, value depends on neither the assumed interaction nor concentration. Each R, value defines a microscopic parameter, C(‘) = ( R$/T~~, for each assumed interaction
Time
(psec)
Fig. 3. Experimental curve and MC-multipolar simulations of the fluorescence decay of the ‘Fs,, state of neodymium ions in Nd(l%),Et(l5%):YAG. The solid curve is a MC simulation for a d-d interaction between D-A pairs. The MC dashed and dot-dashed curves ate. for the case when a q-q and a d-d interaction compete to each other according to the energy transfer constant calculated with Q. (4). The transfer rate. for the dashed curve was up to the first D-A neighbor pairs, and for the dot-dashed was up to the third D-A neighbor pairs.
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(psec)
Time
(psec)
Fig. 4. Experimental curve and MC-multipolar simulations of the fluorescence decay of the 4F,,, state of neodymium ions in Nd(l%),Er(30%):YAG. The solid curve is a MC simulation for a d-d interaction between D-A pairs. The MC dashed curve is for the case when a q-q and a d-d interaction compete to each other according to the energy transfer rate constant calculated with Eq. (4). The transfer rate for the dashed curve was for all D-A pairs. I
0.1 . S . 0 z 2 L z
0.01
0.001
0.0001 loo
Time
ot,sec)
Fig. 5. In this figure we compare the FD and MC predictions, both with a dipole-dipole interaction between D-A pairs, for the fluorescence decay of the 4F,,, state of neodymium ions in Nd,Ho:YAG and Nd,Er:YAG. Curves (a), (b) and (c) correspond to 30% Er, 15% Er, and 10% Ho, respectively, and for constant 1% Nd. The solid curves are. the MC simulations and correspond to the solid curves in Figs. 2-4. The dashed curves are the FD results calculated from Eq. (5). Both models were run with the same parameters for a d-d interaction between D-A pairs.
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between dopants. The corresponding C’“’ values for these R, values are given in Table 1. The used duo value was 240 ps [41. Figs. 2-4 show the experimental and the MC simulations of the Nd+3 decay curves in Nd(1 %), Ho(lO%):YAG, Nd(l%),Er(lS%):YAG and Nd(l%),Er(30%):YAG, respectively. Each figure has an inset for the transients at very short times up to 4 ps, where the vertical axis is in natural scale. For each figure the solid curves correspond to MC simulations for the D emission transient with a d-d interaction between D-A pairs - the C@’ values reported in Table 1 were used. In Fig. 2 the dashed curve is the D emission transient when a d-q and a d-d interaction were assumed to compete. At higher A concen~ations the assumed competing interactions were a q-q and a d-d, and the D emission transient is given by the dashed curves in Figs. 3 and 4.
4. Discussion As expected, the solid curves in Figs. 2-4 do not follow the fast initial decay shown by the experimental curves. However, they follow the experimental decay at long times. This is so because for large D-A distances the d-d interaction overcomes any other shorter range interaction. We also calculate the transients of the three luminescent systems shown in Figs. 2-4 with the FD model for a d-d interaction. In Fig. 5 we compare, in a convenient time scale, the MC and FD results where the solid and the dashed curves correspond to the MC and FD results, respectively. Both models were run with the same R, vflues reported for Figs. 2-4, that is, 9.12 and 8.65 A for Nd,Ho:YAG and Nd,Er:YAG, respectively. The ED D-emission transients were calculated from &;(r) = exp[ t/rDo - $C,( * 3C(6)t) “2] ,
(5)
where, C, is the A concentration. As shown in Fig. 5, the FD decay curve for the luminescent system with lower A concen~ation is very similar to the MC decay trend curve. That is, for 10% Ho+~ the ED decay curve overlaps the MC decay curve starting at about 20.0 IJS and oscillates around the MC curve
279
with very small amplitudes at very long times. For Erf3 at 15 and 30% the overlap of the FD transient to the MC transient is for a short period of time. In other words, Fig. 5 shows that as the A concentration increases the FD transients are less alike to the MC simulations. This result was expected since the FD model was developed for low A concentrations, thus, at very low A concen~ations the ED results are more alike to the MC simulations. At very short times the FD transients show a faster decay than the shown by the d-d MC transients, see the inset of Fig. 5. When the MC model considered a higher order multipolar interaction competing to the d-d interaction, the resulting D emission gave a much better approach to the experimental curves. For each excited D, the higher order multipolar interaction was competing to the d-d interaction according to the transfer rates calculated from Eq. (4). The best approach to the ex~~mental curves at all times was obtained for the system with 15% Er+3, see Fig. 3. The dashed curve in Fig. 3 corresponds to the case of a q-q interaction com~ting against the d-d interaction. As expected, at the beginning of the transient the dashed curve shows a faster decay than the decay shown by the solid curve. This is so since the energy transfer rate due to the q-q interaction has a larger value. That is, the q-q interaction overcomes the d-d at short distances, and this effect is observed as the fast initial decay shown by the dashed curve. At early times, the decay of the dashed curve is slightly faster than the shown by the experimental curve, see inset. At about 3.0 t.i,sthe experimental and the dashed curves overlap; and at longer times the former shows a slightly faster decay. Note that about 3.0 t~.s the solid and the dashed curves also start to overlap each other. Nevertheless, note that at very long times the solid and the dashed curves overlap each other and follow the decay trend of the experimental curve. This is expected since at long times the interaction between D-A pairs is given by the interaction with larger range, i.e., d-d. The energy transfer rate due to the q-q interaction for the dashed curve in Fig. 3 was calculated up to first D-A neighbor pairs. (The transfer rate due to the d-d interaction was calculated for all D-A pairs.) Some differences can be observed in the D emission transients if additional neighboring pairs are considered. For the transfer rate up to the third D-A
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neighbor pair these differences are shown by the dot-dashed curve in Fig. 3. The decay trend is faster than the shown by the experimental curve. It is about 50.0 p.s when it overlaps the other two MC curves shown in Fig. 3. This result is expected: as the range of the q-q interaction is extended the effect in the D emission curve is reflected as a faster initial decay that lasts longer. The small differences between the dashed and experimental curves in Fig. 3 can be attributed to the fact that the MC model does not consider losses at all. When a D ion is excited it can only transfer or emit the excitation energy. Furthermore, any other process or effect that could interfere the non-radiative energy transfer from D to A ions is neglected by the MC model. For the case of Ho+~ at lo%, the luminescent system with lowest A concentration, Fig. 1 shows that the number of first D-A neighbor pairs is low and a d-q interaction was assumed. Furthermore, the experimental fast initial decay shown in Fig. 2 is softer than the observed in Fig. 3. The dashed curve in Fig. 2 shows our MC simulation. In the inset of this figure the dashed curve follows the fast initial decay given by the ex~~mental curve but then gives a faster decay. At long times both MC curves, the solid and dashed, overlap and follow the decay trend shown by the ex~~rnen~l Nd+3 decay. However, note that the experimental curve shows a slightly faster decay between about 10.0 to 30.0 ps. For the luminescent system with the highest A concentration, Er+3 at 30%, a q-q interaction was required to compete against to the d-d interaction. For this luminescent system the transfer rate was calculated for all D-A pairs. The MC D-emission transient is given by the dashed curve in Fig. 4. Again, the MC simulation curve is below the experimental curve at very short time, see inset in Fig. 4. It is at about 7.0 ~1s when the experimental curve is noisy and it is difficult to tell its decay trend at much longer times. The MC simulations predict that the dashed and solid curves will overlap at about 50.0 ps. That is, it is predicted that the experimental
curve is mainly due to the effect of a q-q interaction between the D-A pairs. If the experimental data could be measured for longer times and with more precision for lower order of decays, then a direct effect of the d-d interaction could be observed. The best fit to the experimental curves in Figs. 2-4 is given by the dashed curves. This is not an excellent fit as other fits reported in the literature for other luminescent systems and other models [ 11,121. Nevertheless, the dashed curves follow the decay trend of the measured decays. We did not expect an excellent fit to the experimental curves with the MC model simulations but a fit that could follow their decay trend. We claim that the MC model simulates the process of the non radiative energy transfer between the 24c sites of the YAG lattice, from D to A ions. For our simulations other processes or crystal defects that could interfere with the transfer process were not considered. One important effect that was neglected in our model is due to the variation of the reported nominal concentration of dopants. For a luminescent laser crystal system, as the A concentration increases the faster the depopulation of the excited state of D ions. From Fig. 1 one expects that for higher A concen~ations the initial decay of the D emission transient should become faster due to the increased number of first neighbors. That is, not all D ions have the same dis~ibution of A ions and those with a larger number of A ions, more than the average, will dominate the decay. For these D ions, the D-A distances are shorter and the shorter range multipolar interaction will be enhanced. Although this prediction has been suggested by some authors f7,8] our MC simulations strongly support it by following the decay trends of the experimental curves (Figs. 2-4) reported in Ref. [4]. The exchange interaction was also considered in our MC simulations. For this interaction Eq. (2) holds and the transfer rate, W,,(Rjj), in Eq. (3) for the effective lifetime of D ions was changed. That is, WDA(Rij) is the sum of the two terms, one due to the exchange interaction and the other due to the d-d interaction. The transfer rate for an exchange interac-
Fig. 6. Experimental curve, MC-muitipol~ and MC-exchange simulations of the fluorescence decay of the 4F,,, state of neodymium ions in Nd,Ho:YAG and Nd,Er:YAG with A concentrations of (a) Ho (lo%), (b) Er (15%) and (c) Er (30%). the D concentration was constant, Nd (1%). For all figures the dotted curves correspond to the MC-exchange simulations, see text, and the dashed curves correspond to the MC-muiti~lar simulations, see Figs. 2-5.
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20
j0
40
Time (psec) 1
2
0.1
z! z u) 55 0.01
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20 Time (MSec)
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tion is given by [3,13]
where the sum is over the number of A ions, NA, L is the “effective Bohr radius” and y= ZR,/L is known as the strength of the interaction [4]. The used Ro and Q-no values in the MC simulations in Eq. (6) were the same as those considered for the multipolar cases. Thus, the only unknown parameter in Eq. (6) is L and its value was determined from the simulations. Figs. 6a-6c show the MC simulations and the experimental decay of the 4F,/, state of neodymium ions for the considered luminescent systems, i.e., for polo%), Er(15%) and Er(30%) in YAG, respectively. The dashed curves are for the multipolar interaction according to the above discussion, see Figs. 2-4, and the dotted curves are for a mixture of an exchange and a d-d interaction. For Figs. 6a and 6b the exchange interaction was assumed to be up to first D-A pairs and the determined L values were 1.48 A and L = 1.12 A, respectively. For the luminescent system with Er at 30%, similarly for the case of Er at 15%, we took L = 1.12 A and the exchange interaction was considered up to the third D-A neighbor pair. For the luminescent systems with Ho(lO%) and Er(15%) the dotted curves in Figs. 6a and 6b show an approach very similar to the reported with the MC simulations with multipolar interactions, dashed curves. (Note that both simulation curves almost overlap to each other and give a good approach to the experimental decay.) However, for Eri30%) the approach of the exchange interaction and d-d interaction, dotted curve in Fig. 6c, to the experimental curve is not as good as the shown by the dashed. The faster decay at early times shown by the dotted curve compared to the ex~~mental decay in Fig. 6c shows that the exchange interaction between neighboring D-A is much stronger, as expected, than the q-q interaction - dashed curve. Note, however, that for the other two luminescent systems with lower A concen~ations, Figs. Ba, 6b, both MC simulations give a very good approach to the experimen-
Communications 129 (1996) 273-283
tal curves. That is, as the A concen~ation increases the exchange interaction becomes stronger than the multipolar interaction (e.g., d-q or q-q as considered). This result is shown by our MC simulations and the MC model predicts that the interaction between neighboring D-A pairs in Nd,Ho:YAG and Nd,Er:YAG is multipolar. The GKS model used in Ref. [4] for the nonradiative energy transfer for the three luminescent systems also assumed an exchange and a d-d interaction. Nevertheless, for the exchange interaction it is not reported the values for R, nor for L or y, For our simulations we used the same R, values found for the multiply case which are more similar than those reported in Ref. 141. 5. Conclusions It is found that the non-radiative energy transfer from Nd’3 to Ho’~ and from Nd*3 to Erf3 in codoped Nd,Ho:YAG and Nd,Er:YAG is driven by two multipolar interactions. For Nd-Ho pairs the interactions are a d-q and a d-d, and for Nd-Er pairs they are a q-q and a d-d. These interactions compete to each other according to their range and A concentrations. The MC simulations show that the number of nearest D-A neighbor pairs is increased as the A concen~ation increases. The effects and range of higher order multipolar interactions are enhanced when the A concentration is increased, and are reflected on the measured fast initial Nd+3 decay. Our MC simulations describe the measured Nd+3 decay trends reported experimentally. Acknowledgements This work was partially supported by CONACyT under grant 3245-E and with a doctoral scholarship for E.J.I. OBG gratefully acknowledge the help and technical discussions regarding the early stage of the development of the MC model with B. Di Bartolo. References [I ] A.A. Kaminskii, Laser Crystals (Springer, Berlin, 1981). 121 T. Forster, ANI. F’hys. 2 (1948) 55. [3] D.L. Dexter, J. Chem. 21 (1953) 836. [4] M. Stalder and M. Bass, J. Opt. Sot. Am. B 8 (1991) 177.
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[5] S.I. Golubov and Yu.V. Konobeev. Fiz. Tverd. Tela (Leningrad) 13 (1971) 3185 [Sov. Phys. Solid State 13 ( 1972) 26791. [6] V.P. Saktm, Fiz. Tverd. Tela (Leningrad) 14 (1971) 2199. [Sov. Phys. Solid State 14 (1973) 19061. [7] J.D. Axe and P.F. Weller, J. Chem. Phys. 40 (1964) 3006. [S] L.A. Riseberg and M.J. Weber, in: Progress in Optics, Vol. XIV, ed. E. Wolf (North-Holland, Amsterdam, 1976) pp. 89157.
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and C.W. Struck, J. Chem. Phys. 100 [910. Barbosa-Garcia (1994) 4554. t101 R.W.G. Wyckoff, Crystal Structures, Vol. 3, 2nd Ed. (Interscience Publishers, Wiley, New York, 1965). [Ill I. Lupei, A. Lupei, C. Georgescu and C. Ionescu, Optics Comm. 60 (1987) 59. [121S.R. Rotman, Appl. Phys. Len. 54 (1989) 2053. [131N. Bodensachatz, R. Wamtemacher, J. Heber and D. Mateika, J. Lumin. 47 (1991) 159.
For t?xw luminese~nt laser ~~s~~s it is fmmd that the d~~~~-dj~o~e jntem~~~~nbetween d~n~r-a~~e~t~r pairs at short distances is damina~d by hewer order multipa~ar ~ntemetions. This effect is refiected as a fast initial decay in the donor emission ~sient of the codoped systems Nd,Ha:YAG and Nd,ErzYAG at high acceptor ~~n~en~ti~ns. The effect is enhanced as the acce@or c~cen~~~n is ‘ii~~~sed, At iong times the donor emission transients show tile effects of the dip&e-dipole interaction between ~~~~~-acce~~~rpairs, that is, the long range multipolar interaction drives the non-radiative energy transfer, The Monte Carlo model shows these effects for the non-radiative energy transfer process applied to those luminescent systems. For this modei the discreteness of the YAG lattice is kept since dopants are randomly placed into the 24c sites. The donor emission ~ns~~~~ are obtained by calculating the ~~~a~~i~t~es of the events that accur after a donor is excited.
The RaR~radiati~eeReTgy transfer pracess has played a major rofe in jnc~~~n~ t&e ~urnpj~~ efficiency of active ions in. soEd state hX%rmatetials ffb An ~~te~~~t~o~between two ions, one in its ground state (the zceptor A) and the other in some excited state (the donor D), drives this process. As a result of the process the excitation moves from ions D to A. Two specific non-radiative isotropic D-A interactions have been considered, namely, rnu~tip~l~ and exchange. The first is an ~nt~rac~~n d~c~m~sabIe into dj~Ie-d~~~~ Id-d), ~~~e-qu~dmpol~ (d-q), qua~p~~e-quadm~~~
The mode1 developed by Forster and Dexter [2J] CFDI for the nob-r~iat~~e energy transfer considers t&at dopants are ~~d~rn~~ placed in the frost material ~eve~~~ess~ f@r tfie ~a~~jca~ expression of this model a cantinuous dist~bution of A ions is assumed, i-e,, for the case of laser crystal rnat~~~a~s the discreteness of tke Mice is neglected. For I& model it is not possible to consider a mixture of interactions between the interacting B-A pairs. FLITthermore, this madei was developed for fow dopant concen~atjons, where D-A average distances ;fpe? large. Despite these drawbacks the FD mode1 has
0030-40~1/96/$IZ,W copyright 0 I %Xi EIsevier Science I3.V”Aft rights reserved. Pff s0030-40 18(96)00062-4
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found wide use for the analysis of experimental data in laser crystals since it gives the basic physics of the transfer process. With a fast detection system Stalder and Bass [4] were able to measure a fast initial decay of the 4F,,, state of neodymium ions in three luminescent systems: Nd(l%),Ho(lO%):YAG, Nd(l%), Er(lS%):YAG and Nd(l%),Er(30%):YAG. For these systems the D and A ions are Nd+3 and Erf3 or Ho+~, respectively. As expected, the FD model did not give a complete understanding of the kind of interaction that was driving the transfer. They turned to the model developed by Golubov-KonobeevSakun (GKS) [5,6] for the non-radiative energy transfer. It was reported that the neodymium fast initial decay was due to a short-range interaction involving the nearest-neighbor sites. As a result they assumed an exchange interaction for short D-A distances and for further distances a d-d interaction. The fraction of D ions that decayed during the fast initial decay was determined from the experiment and with the GKS model. Thus, they concluded that A ions placed in the four nearest-neighbor sites were responsible for the fast initial decay. It is known that d-d interactions can generally be expected to dominate in systems ch~acteri~d by allowed electric dipole radiative transitions [3]. Axe and Weller [7] have pointed out that, although the radiative transitions within the 4f” rare earth configurations are generally electric dipole in nature they are much weaker than fuIly allowed dipole transitions. As a result quad~pol~ processes become of greater importance and can completely dominate non-radiative transfer processes in certain systems. This is more probable to occur with luminescent systems at high A concentrations where the D-A separation is small [S]. The experimental Nd*3 decay curves measured by Stalder and Bass seem to be a clear example where high order multipolar interactions between D-A pairs dominate the d-d interaction at short distances. We support this prediction with Monte Carlo (MC) simulations. In this work those experimental neodymium decay curves are analyzed with the MC model for the non-radiative energy transfer process [9]. For the luminescent systems, Nd(l%),Er(15%3):YAG and Nd(l%),Er(30%):YAG, it is found that the energy transfer is driven by two multipolar interactions, a
Communicutions
129 11996) 273-283
q(R)-q(R) and a d(R)-d(R). For the case of Nd( 1%),Ho( IO%):YAG the driving interactions are a d(R)-q(R) and a d(R)-d(R). For each luminescent system the interactions compete to each other, and the stronger and shorter range interaction dominates the d(R)-d(R) interaction for short D-A distances. This effect is reflected as the fast initial Nd+3 decay measured experimentally, that is expected in the MC simulations. At large D-A distances the d-d interaction dominates the transfer process and the D emission transients calculated by the MC model also follow the experimental decay trends at long times. These effects are enhanced as the A concentration increases.
2. The Monte Carlo (MC) model The MC model for the non-radiative energy transfer process in laser crystals has been reported elsewhere [9], and in here we give a brief account of it. The crystal structure of YAG is 1a3d (230) and the site positions of their components are well known [IO]. When this crystal is doped with rare earth elements they substitute the yttrium ions at 24~ sites. This model considers a cube containing 1000 unit cells of YAG as a working unit that is replicated to fill all space. By using the MC method dopants are randomly placed in the potential crystal sites of this cubic crystal sample. Afterwards they are randomly labeled as D and A ions. With this random placement each D ion has a particular dis~bution of A ions. An interaction between D-A pairs is assumed and the probabilities of the events that occur after a D ion is excited are calculated, i.e., the D and A emission transients. For this work we run 192000 excitations for each codoped crystal. That is, the MC model results, for each sample crystal, are the averages of 800 codoped crystals and exciting each D ion at t=O. For a Coulomb multipole interaction between D-A pairs the transfer rate between the &h-D ion and the jth-A ion, separated by Rlj, is defined by
WDA(Rij)
=
(1)
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Distribution
I
I
I
I
’
I
I
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probability of nearest neighbors YAG: D*‘, AC3 I
/
‘I
/
/,‘,I/,‘//,,
0.8
x
R31%
l
A”
10% A'" 15%
l
A*‘30%
l
X
l
II
x
xx
X
x
X=Rda/Rmin Fig. 1. Probability distribution of nearest neighbors at four A concentrations in codoped D+‘,A+‘:YAG. (D+$ and A+’ stand for trivalent donor and acceptor ions for Nd+3 and Hof3 or Er+3, respectively.) Dopants were randomly placed into 24~ sites of the YAG lattice and afterwards were randomly labeled as D and A ions. The D ions were fixed at 1% concentration and A ions at 1, 10, 15 and 30%. The vertical axis is the probability PD,,, (X,) of finding the nearest D-A distance between X, and X, + 0.2, where X, is adimensional and defined as the ratio between the nearest D-A pair distance RDA,n and the minimum distance between the 24~ sites. The scatter of the plot is due entirely to the discreteness of the lattice.
where Cfsf are the microscopic interaction parameters and the values s = 6, 8, 10,. . . correspond to a d-d, d-q, q-q,. . . interactions, respectively. The microscopic interaction parameter for a particular interaction is defined as Cfs) = (ROS)S/rnO, where R,, is the critical transfer distance defined as the separation between D-A pairs for which the energy transfer rate is equal to (r&Y’, and in,, is the D
life time when A ions are not present. Note that the MC model can consider a single multipolar interaction, as the FD model does, or a mixture of interactions to drive the energy transfer. Once dopants are randomly placed, one can calculate any of the terms in IQ. (1) or assume any mixture between them. To calculate any of the terms in Eq. (1) one has to look for the best R,, value to fit the experimental decay
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Table 1 C”’ values used in the MC simulations for the fluorescence decay of the 4F,,, state of neodymium ions in Nd(l%),Ho(lO%):YAG, Nd(1 %),Erl30%):YAG and Nd(l %),E1(30%):YAG. The C(I) values are defined by (R,)“/T,~, where R, = 9.12 and 8.65 i for the systems with holmium and erbium, respectively, and rot = 240 t.~s. Note that the R, value for holmium is constant for the competing interactions d-q and q-q. For erbium the R, value is constant for the competing interactions d-d and q-q and for both concentrations. The units for C’“’ values am in cm’/s
~(6) c(8) c(lO)
HdlO%)
Ert15%)
Ert30%)
2.39x 10-39 1.99x 10-53 _
1.75x 10-39 -
1.75x
0.98x
0.98X 1O-67
1O-67
10-39
curve. Clearly if a mixture of interactions is needed a single R,, value should be found for all interactions. The total emission transient will be the individual emission transient, averaged over the D ions which are excited at t = 0, namely 4,(t)
= (exp[-t/(T,ff)i])Av.overi,
(2)
where (rerrji is the effective lifetime for the ith
1
_
129 (1996) 273-283
donor, defined by 1 (‘eff)i
1 = -
+ 2 W,,( R,J
'DO
j=l
(3)
where the sum is over the number of A ions, NA,and s = 3, 4, 5,. . . for the multipolar interaction between D-A pairs. The average of Eq. (2), i.e., the averaging of the responses of many excited donors each having its own (T~~~)~, is done through probability expressions of the corresponding emissions. As shown in Ref. [9] the average of Eq. (2) is non-exponential. This behavior is a direct result of the average procedure and of the assumed interactions between D-A pairs. If the interaction between D-A pairs is neglected, the second term in Eq. (3) is zero and Eq. (2) gives an exponential behavior, as expected [9].
I-
0.1
Zl 0’ 3 Z Z 5 0.01
0.001
0
40
80 Time
120
(psec)
Fig. 2. Experimental curve and MC-multipolar simulations of the fluorescence decay of the 4Fs,2 state of neodymium ions in Nd(l%),HdlO%):YAG at long and short times. Horizontal axis is time in psec. The vertical axis is intensity in au. showing three orders of decays at long times, and the inset is in natural scale. The solid curve corresponds to a MC simulation for a d-d interaction between D-A pairs. The MC dashed curve is for the case when a d-q and a d-d interaction compete to each other according to the energy transfer rate constant calculated with Eq. (4). The transfer rate for the d-q interaction was calculated up to the first D-A neighbor pairs.
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21-l
the concentration of A ions. Conversely, as the A concentration decreases, e.g., A ions at l%, there is an increasing probability that no A ions exist at small values of X, and that the next D-A pair is at some large X, value. Thus, in the case of A ions at 15%, Fig. 1 shows that almost 50% of D ions have at least an A ion as first neighbor, nearly 40% have an A ion as second neighbor, and so on. For a particular multipolar interaction between D-A pairs, the energy transfer rate was calculated according to NAP(S)
3. Results
Once the YAG crystal is codoped, each D ion has a particular distribution of A ions, and the distribution probability of nearest D-A pairs is calculated. In other words, after codoping the crystal each D ion is inspected for the distance to its nearest A ion, R DA.n. This distante is made non-dimensional by dividing it by 3.67 A - the shortezt distance between 24c sites; thus, X, = R,,,,/3.67 A. The RI)*,” value is counted in its own bin between X, and X, + 0.2, with X, incrementally increasing from 0.0 to 9.8. Fig. 1 shows this probability distribution for nearest neighbors, PDA,n(Xn), at four A ion concentrations, l%, lo%, 15% and 30%. The D ion concentration is fixed at 1%. The functions PDA,n(Xn) shown in Fig. 1 are not continuous because of the discreteness of the YAG lattice. The two zero probabilities at X, = 1.2 and 1.4 correspond to distances which are not present in the lattice. Also, the figure shows an increment in the probability of nearest D-A neighbor pairs with
(4) where the sum is over NA, the total number of A ions. The critical transfer distance, R,, = R,, is the only free parameter for the MC model and it was constant and unique, 9.12 and 8.65 A for Nd,Ho:YAG and Nd, Er:YAG, respectively. The R, value depends on neither the assumed interaction nor concentration. Each R, value defines a microscopic parameter, C(‘) = ( R$/T~~, for each assumed interaction
Time
(psec)
Fig. 3. Experimental curve and MC-multipolar simulations of the fluorescence decay of the ‘Fs,, state of neodymium ions in Nd(l%),Et(l5%):YAG. The solid curve is a MC simulation for a d-d interaction between D-A pairs. The MC dashed and dot-dashed curves ate. for the case when a q-q and a d-d interaction compete to each other according to the energy transfer constant calculated with Q. (4). The transfer rate. for the dashed curve was up to the first D-A neighbor pairs, and for the dot-dashed was up to the third D-A neighbor pairs.
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(psec)
Time
(psec)
Fig. 4. Experimental curve and MC-multipolar simulations of the fluorescence decay of the 4F,,, state of neodymium ions in Nd(l%),Er(30%):YAG. The solid curve is a MC simulation for a d-d interaction between D-A pairs. The MC dashed curve is for the case when a q-q and a d-d interaction compete to each other according to the energy transfer rate constant calculated with Eq. (4). The transfer rate for the dashed curve was for all D-A pairs. I
0.1 . S . 0 z 2 L z
0.01
0.001
0.0001 loo
Time
ot,sec)
Fig. 5. In this figure we compare the FD and MC predictions, both with a dipole-dipole interaction between D-A pairs, for the fluorescence decay of the 4F,,, state of neodymium ions in Nd,Ho:YAG and Nd,Er:YAG. Curves (a), (b) and (c) correspond to 30% Er, 15% Er, and 10% Ho, respectively, and for constant 1% Nd. The solid curves are. the MC simulations and correspond to the solid curves in Figs. 2-4. The dashed curves are the FD results calculated from Eq. (5). Both models were run with the same parameters for a d-d interaction between D-A pairs.
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between dopants. The corresponding C’“’ values for these R, values are given in Table 1. The used duo value was 240 ps [41. Figs. 2-4 show the experimental and the MC simulations of the Nd+3 decay curves in Nd(1 %), Ho(lO%):YAG, Nd(l%),Er(lS%):YAG and Nd(l%),Er(30%):YAG, respectively. Each figure has an inset for the transients at very short times up to 4 ps, where the vertical axis is in natural scale. For each figure the solid curves correspond to MC simulations for the D emission transient with a d-d interaction between D-A pairs - the C@’ values reported in Table 1 were used. In Fig. 2 the dashed curve is the D emission transient when a d-q and a d-d interaction were assumed to compete. At higher A concen~ations the assumed competing interactions were a q-q and a d-d, and the D emission transient is given by the dashed curves in Figs. 3 and 4.
4. Discussion As expected, the solid curves in Figs. 2-4 do not follow the fast initial decay shown by the experimental curves. However, they follow the experimental decay at long times. This is so because for large D-A distances the d-d interaction overcomes any other shorter range interaction. We also calculate the transients of the three luminescent systems shown in Figs. 2-4 with the FD model for a d-d interaction. In Fig. 5 we compare, in a convenient time scale, the MC and FD results where the solid and the dashed curves correspond to the MC and FD results, respectively. Both models were run with the same R, vflues reported for Figs. 2-4, that is, 9.12 and 8.65 A for Nd,Ho:YAG and Nd,Er:YAG, respectively. The ED D-emission transients were calculated from &;(r) = exp[ t/rDo - $C,( * 3C(6)t) “2] ,
(5)
where, C, is the A concentration. As shown in Fig. 5, the FD decay curve for the luminescent system with lower A concen~ation is very similar to the MC decay trend curve. That is, for 10% Ho+~ the ED decay curve overlaps the MC decay curve starting at about 20.0 IJS and oscillates around the MC curve
279
with very small amplitudes at very long times. For Erf3 at 15 and 30% the overlap of the FD transient to the MC transient is for a short period of time. In other words, Fig. 5 shows that as the A concentration increases the FD transients are less alike to the MC simulations. This result was expected since the FD model was developed for low A concentrations, thus, at very low A concen~ations the ED results are more alike to the MC simulations. At very short times the FD transients show a faster decay than the shown by the d-d MC transients, see the inset of Fig. 5. When the MC model considered a higher order multipolar interaction competing to the d-d interaction, the resulting D emission gave a much better approach to the experimental curves. For each excited D, the higher order multipolar interaction was competing to the d-d interaction according to the transfer rates calculated from Eq. (4). The best approach to the ex~~mental curves at all times was obtained for the system with 15% Er+3, see Fig. 3. The dashed curve in Fig. 3 corresponds to the case of a q-q interaction com~ting against the d-d interaction. As expected, at the beginning of the transient the dashed curve shows a faster decay than the decay shown by the solid curve. This is so since the energy transfer rate due to the q-q interaction has a larger value. That is, the q-q interaction overcomes the d-d at short distances, and this effect is observed as the fast initial decay shown by the dashed curve. At early times, the decay of the dashed curve is slightly faster than the shown by the experimental curve, see inset. At about 3.0 t.i,sthe experimental and the dashed curves overlap; and at longer times the former shows a slightly faster decay. Note that about 3.0 t~.s the solid and the dashed curves also start to overlap each other. Nevertheless, note that at very long times the solid and the dashed curves overlap each other and follow the decay trend of the experimental curve. This is expected since at long times the interaction between D-A pairs is given by the interaction with larger range, i.e., d-d. The energy transfer rate due to the q-q interaction for the dashed curve in Fig. 3 was calculated up to first D-A neighbor pairs. (The transfer rate due to the d-d interaction was calculated for all D-A pairs.) Some differences can be observed in the D emission transients if additional neighboring pairs are considered. For the transfer rate up to the third D-A
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neighbor pair these differences are shown by the dot-dashed curve in Fig. 3. The decay trend is faster than the shown by the experimental curve. It is about 50.0 p.s when it overlaps the other two MC curves shown in Fig. 3. This result is expected: as the range of the q-q interaction is extended the effect in the D emission curve is reflected as a faster initial decay that lasts longer. The small differences between the dashed and experimental curves in Fig. 3 can be attributed to the fact that the MC model does not consider losses at all. When a D ion is excited it can only transfer or emit the excitation energy. Furthermore, any other process or effect that could interfere the non-radiative energy transfer from D to A ions is neglected by the MC model. For the case of Ho+~ at lo%, the luminescent system with lowest A concentration, Fig. 1 shows that the number of first D-A neighbor pairs is low and a d-q interaction was assumed. Furthermore, the experimental fast initial decay shown in Fig. 2 is softer than the observed in Fig. 3. The dashed curve in Fig. 2 shows our MC simulation. In the inset of this figure the dashed curve follows the fast initial decay given by the ex~~mental curve but then gives a faster decay. At long times both MC curves, the solid and dashed, overlap and follow the decay trend shown by the ex~~rnen~l Nd+3 decay. However, note that the experimental curve shows a slightly faster decay between about 10.0 to 30.0 ps. For the luminescent system with the highest A concentration, Er+3 at 30%, a q-q interaction was required to compete against to the d-d interaction. For this luminescent system the transfer rate was calculated for all D-A pairs. The MC D-emission transient is given by the dashed curve in Fig. 4. Again, the MC simulation curve is below the experimental curve at very short time, see inset in Fig. 4. It is at about 7.0 ~1s when the experimental curve is noisy and it is difficult to tell its decay trend at much longer times. The MC simulations predict that the dashed and solid curves will overlap at about 50.0 ps. That is, it is predicted that the experimental
curve is mainly due to the effect of a q-q interaction between the D-A pairs. If the experimental data could be measured for longer times and with more precision for lower order of decays, then a direct effect of the d-d interaction could be observed. The best fit to the experimental curves in Figs. 2-4 is given by the dashed curves. This is not an excellent fit as other fits reported in the literature for other luminescent systems and other models [ 11,121. Nevertheless, the dashed curves follow the decay trend of the measured decays. We did not expect an excellent fit to the experimental curves with the MC model simulations but a fit that could follow their decay trend. We claim that the MC model simulates the process of the non radiative energy transfer between the 24c sites of the YAG lattice, from D to A ions. For our simulations other processes or crystal defects that could interfere with the transfer process were not considered. One important effect that was neglected in our model is due to the variation of the reported nominal concentration of dopants. For a luminescent laser crystal system, as the A concentration increases the faster the depopulation of the excited state of D ions. From Fig. 1 one expects that for higher A concen~ations the initial decay of the D emission transient should become faster due to the increased number of first neighbors. That is, not all D ions have the same dis~ibution of A ions and those with a larger number of A ions, more than the average, will dominate the decay. For these D ions, the D-A distances are shorter and the shorter range multipolar interaction will be enhanced. Although this prediction has been suggested by some authors f7,8] our MC simulations strongly support it by following the decay trends of the experimental curves (Figs. 2-4) reported in Ref. [4]. The exchange interaction was also considered in our MC simulations. For this interaction Eq. (2) holds and the transfer rate, W,,(Rjj), in Eq. (3) for the effective lifetime of D ions was changed. That is, WDA(Rij) is the sum of the two terms, one due to the exchange interaction and the other due to the d-d interaction. The transfer rate for an exchange interac-
Fig. 6. Experimental curve, MC-muitipol~ and MC-exchange simulations of the fluorescence decay of the 4F,,, state of neodymium ions in Nd,Ho:YAG and Nd,Er:YAG with A concentrations of (a) Ho (lo%), (b) Er (15%) and (c) Er (30%). the D concentration was constant, Nd (1%). For all figures the dotted curves correspond to the MC-exchange simulations, see text, and the dashed curves correspond to the MC-muiti~lar simulations, see Figs. 2-5.
0. Barbosa-Garcia er al. / Optics Communications 129 (1996) 273-283
20
j0
40
Time (psec) 1
2
0.1
z! z u) 55 0.01
(psec)
20 Time (MSec)
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tion is given by [3,13]
where the sum is over the number of A ions, NA, L is the “effective Bohr radius” and y= ZR,/L is known as the strength of the interaction [4]. The used Ro and Q-no values in the MC simulations in Eq. (6) were the same as those considered for the multipolar cases. Thus, the only unknown parameter in Eq. (6) is L and its value was determined from the simulations. Figs. 6a-6c show the MC simulations and the experimental decay of the 4F,/, state of neodymium ions for the considered luminescent systems, i.e., for polo%), Er(15%) and Er(30%) in YAG, respectively. The dashed curves are for the multipolar interaction according to the above discussion, see Figs. 2-4, and the dotted curves are for a mixture of an exchange and a d-d interaction. For Figs. 6a and 6b the exchange interaction was assumed to be up to first D-A pairs and the determined L values were 1.48 A and L = 1.12 A, respectively. For the luminescent system with Er at 30%, similarly for the case of Er at 15%, we took L = 1.12 A and the exchange interaction was considered up to the third D-A neighbor pair. For the luminescent systems with Ho(lO%) and Er(15%) the dotted curves in Figs. 6a and 6b show an approach very similar to the reported with the MC simulations with multipolar interactions, dashed curves. (Note that both simulation curves almost overlap to each other and give a good approach to the experimental decay.) However, for Eri30%) the approach of the exchange interaction and d-d interaction, dotted curve in Fig. 6c, to the experimental curve is not as good as the shown by the dashed. The faster decay at early times shown by the dotted curve compared to the ex~~mental decay in Fig. 6c shows that the exchange interaction between neighboring D-A is much stronger, as expected, than the q-q interaction - dashed curve. Note, however, that for the other two luminescent systems with lower A concen~ations, Figs. Ba, 6b, both MC simulations give a very good approach to the experimen-
Communications 129 (1996) 273-283
tal curves. That is, as the A concen~ation increases the exchange interaction becomes stronger than the multipolar interaction (e.g., d-q or q-q as considered). This result is shown by our MC simulations and the MC model predicts that the interaction between neighboring D-A pairs in Nd,Ho:YAG and Nd,Er:YAG is multipolar. The GKS model used in Ref. [4] for the nonradiative energy transfer for the three luminescent systems also assumed an exchange and a d-d interaction. Nevertheless, for the exchange interaction it is not reported the values for R, nor for L or y, For our simulations we used the same R, values found for the multiply case which are more similar than those reported in Ref. 141. 5. Conclusions It is found that the non-radiative energy transfer from Nd’3 to Ho’~ and from Nd*3 to Erf3 in codoped Nd,Ho:YAG and Nd,Er:YAG is driven by two multipolar interactions. For Nd-Ho pairs the interactions are a d-q and a d-d, and for Nd-Er pairs they are a q-q and a d-d. These interactions compete to each other according to their range and A concentrations. The MC simulations show that the number of nearest D-A neighbor pairs is increased as the A concen~ation increases. The effects and range of higher order multipolar interactions are enhanced when the A concentration is increased, and are reflected on the measured fast initial Nd+3 decay. Our MC simulations describe the measured Nd+3 decay trends reported experimentally. Acknowledgements This work was partially supported by CONACyT under grant 3245-E and with a doctoral scholarship for E.J.I. OBG gratefully acknowledge the help and technical discussions regarding the early stage of the development of the MC model with B. Di Bartolo. References [I ] A.A. Kaminskii, Laser Crystals (Springer, Berlin, 1981). 121 T. Forster, ANI. F’hys. 2 (1948) 55. [3] D.L. Dexter, J. Chem. 21 (1953) 836. [4] M. Stalder and M. Bass, J. Opt. Sot. Am. B 8 (1991) 177.
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[5] S.I. Golubov and Yu.V. Konobeev. Fiz. Tverd. Tela (Leningrad) 13 (1971) 3185 [Sov. Phys. Solid State 13 ( 1972) 26791. [6] V.P. Saktm, Fiz. Tverd. Tela (Leningrad) 14 (1971) 2199. [Sov. Phys. Solid State 14 (1973) 19061. [7] J.D. Axe and P.F. Weller, J. Chem. Phys. 40 (1964) 3006. [S] L.A. Riseberg and M.J. Weber, in: Progress in Optics, Vol. XIV, ed. E. Wolf (North-Holland, Amsterdam, 1976) pp. 89157.
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and C.W. Struck, J. Chem. Phys. 100 [910. Barbosa-Garcia (1994) 4554. t101 R.W.G. Wyckoff, Crystal Structures, Vol. 3, 2nd Ed. (Interscience Publishers, Wiley, New York, 1965). [Ill I. Lupei, A. Lupei, C. Georgescu and C. Ionescu, Optics Comm. 60 (1987) 59. [121S.R. Rotman, Appl. Phys. Len. 54 (1989) 2053. [131N. Bodensachatz, R. Wamtemacher, J. Heber and D. Mateika, J. Lumin. 47 (1991) 159.