Up-Conversion and Migration by Energy ... - American Chemical Society

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Up-Conversion and Migration by Energy Transfer: A Mixed Model for Doped Luminescent Solids F. Ferraro‡ and C. Z. Hadad*,† †

Grupo de Química-Física Teórica, Instituto de Química, Universidad de Antioquia, Calle 67 No. 53·108, bloque 2 oficina 337, Apartado Aéreo 1226, Medellín, Colombia ‡ Departamento de Ciencias Químicas, Facultad de Ciencias Exactas, Universidad Andrés Bello, Av. República 275, Santiago de Chile, Chile ABSTRACT: From a fundamental and technological perspective energy transfer up-conversion in luminescent solids is a very important phenomenon that requires different viewpoints for understanding and modeling. The efficiency of this process increases in the presence of energy migration. In this work a model to study the energy migration effects on the up-conversion process for doped solid time-dependent luminescence is proposed. The model takes advantage of the Förster-like method to calculate the relevant quantity averages and a microscopic−macroscopic methodology to solve energy transfer mixed schemes. Along with adequately describing the experimental trends reported in the literature for the relevant involved states, the model separates the migration-excited transient population from the initially excited time-dependent population, which permits separate analysis of the dynamics of the components. This last aspect is difficult to achieve experimentally and, to our knowledge, has not been reported previously in the theoretical literature.

■

INTRODUCTION The understanding of energy transfer (ET) processes among optical centers in solid materials is of great interest due to its utility in the manufacture of different kinds of devices.1−10 From the viewpoint of its physical possibility and according to conventional theories, ET processes could take place by electric or magnetic multipole interactions, by exchange interactions, or by emission and subsequent absorption of photons.7,9,10 Also, there are several possible kinetic mechanisms by which ET can occur: cross relaxation, energy migration, back-transfer, and frequency up-conversion, among others.7,9−15 Both the nature of the host lattice and the nature of the optical centers are determinantal in the preferred kinetic mechanism. However, very frequently it is possible to find two or more mechanisms operating simultaneously.10,16−21 One of the most relevant ET kinetic mechanisms and one that has received special attention in the past few years is energy transfer up-conversion, ETU.22−27 In this case energy can be transferred by one or more initially excited optical centers to other centers also in some excited state, generating higher excited states, which can emit radiation of shorter wavelength than would be possible from the initially excited state. This effect allows applications such as display devices,4,5 biosensors,3,28 up-conversion lasers,29−31 and other devices.10,32 Usually, the ETU efficiency improves in the presence of energy migration, ETM (energy transfer migration).18−21,33−36 This is because during the migration the energy is transferred from one excited optical center to another center in the ground © 2012 American Chemical Society

state, lengthening the lifetime of the excited state and increasing the probability of the up-conversion phenomenon from this state. Not only because of its everyday increasing technological uses, but also because of its simplicity from a theoretical point of view, especially attractive is ETU between trivalent rare earth ions in monodoped solids.15,37−44 Likewise, among the possible studies related to ETU among doped solids, the analysis of the time-resolved luminescence from the involved states plays an important role in the inquiry of the up-conversion mechanisms, the kind of interactions, its magnitude, and considerations about the dynamics of the process.2,7,9,15,18,19,33−36 On the other hand, it is widely recognized that the Försterlike method to describe the time-dependent luminescence of the ET-affected quenched state45,46 is a correct procedure for solids with randomly distributed optical centers, because it has been amply corroborated in various systems, and where it does not work or where it is not directly applicable, the model and the procedure have been successfully extended and adapted to include conditions and cases not considered in the original model.19,46−51 The Förster-like method begins with the differential equation for the dynamics of the optical center microscopic probabilities of being in the state of interest. Those probabilities are affected by the interactions among the optical Received: December 7, 2011 Revised: February 20, 2012 Published: February 24, 2012 7134

dx.doi.org/10.1021/jp211797e | J. Phys. Chem. C 2012, 116, 7134−7143

The Journal of Physical Chemistry C

Article

Figure 1. Relevant energy levels and transitions for the interacting optical centers (see the text for details).

within the excitation time). From there, optical centers have the possibility of intracenter radiative and/or nonradiative decay (descendent arrow/zigzag arrow in the figure), with total rate (2) (2) γr−nr = γr(2) + γnr . The optical centers also have the possibility to transfer their energy to other optical centers that are either in state 2 or in the ground state 0, with interoptical center distance-dependent rates W(rvu) and M(rvu), respectively. In the first case, the result is optical centers in another excited state, labeled 3, of higher energy than the 2 state (up-conversion process), from which up-converted luminescence can be emitted or quenched nonradiatively, with a total intracenter (3) (3) = γr(3) + γnr . In the second case, the result is a rate γr−nr migration of the energy from the initially excited centers to others, enduring with this the effective lifetime of state 2. Microscopic rates of ETU, W(rvu), and ETM, M(rvu), are given by the conventional approximations7,9,10,48−51

centers formalized through the microscopic distance-dependent ET rates. The differential equations are partially solved, and then the Förster-like method is applied to solve the averages for randomly distributed optical centers, which can be found in detail in refs 45, 46, and 49. Likewise, in previous papers49,51 we proposed a “microscopic−macroscopic” methodology to treat mixed and more complex energy transfer schemes to reach the timedependent quantities of interest. We start by outlining the differential equations for the time variation of microscopic probabilities of being in the relevant states and then transforming them into equations for the related macroscopic populations and solve those equations in connection with the resolution of the average of a key quantity for the problem solution. In this study we develop a mixed Förster-like and microscopic− macroscopic analytical model for the ETU phenomenon assisted by energy migration to describe the relevant states time-resolved luminescence of solids monodoped with trivalent lanthanide ions. We found that the model correctly reproduces the literature experimental trends, which shows that one-step migration that incorporates a luminescence retention time is enough to correctly describe the problem, and that by artificially separating the initially excited population from the migration excited population it is possible to reach some understanding about factors that affect the observable population variation and to study the behavior of the components separately and their interrelation.

W (rvu) =

ΩW , s s rvu

and

M(rvu) =

ΩM , s s rvu

(1)

where s = 6, 8, and 10, for dipole−dipole, dipole−quadrupole, and quadrupole−quadrupole interactions, respectively. The terms ΩW,s and ΩM,s are effective interaction constants in this model.49−51 They are representative of any pair of interacting optical centers in the solid (i.e., they are averages among all interacting pairs) and can be expressed in terms of spectroscopic and electronic parameters;50 therefore, they are characteristic of the nature of each optical center−matrix pair. They can also be related to the widely known Förster critical transfer distances, RW and RM,7,9,21,33,34,45 through

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MODEL The diagram of Figure 1 shows the relevant transitions and energy levels involved in our model for the analysis of the mixing between the ETU and ETM. Interacting optical centers (u or v generic centers) are initially found in an intermediate excited state, labeled 2, or in the ground state, labeled 0. The intermediate state 2 is initially reached after pulsed selective excitation of intensity Iexc, short enough not to give rise to significant populations in other excited states (reached by ET

s ΩW , s = γ(2) r−nrRW

and

s ΩM , s = γ(2) r−nrRM

(2)

We can divide the total optical center population contained in a volume V of the solid, NV, into a time-dependent sum of 7135



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optical center populations that remain in any of the relevant possible states 0, 2, and 3 at time t:

dρ(2) u ,III(t ) dt

(2) (2) NV(0)(t ) + NI,(2) V (t ) + NII, V (t ) + NIII, V (t ) + ...

+ NV(3)(t ) = NV

(3)

v=1

+2

dρ(3) u (t ) dt

with u = 1, 2, 3, ..., NV. As can be seen in eq 6 and in the level diagram at the center of Figure 1, the microscopic probability of being in state 2 and in group I for a generic center u decreases with the possibility of intracenter decay, with the possibility of energy transfer to any center v in state 0 (migration), and with the possibility of energy transfer to or from any center v in state 2 of any group, I, II, III, ... (up-conversion). Equation 7 and the level diagram at the right of Figure 1 show that the microscopic probability of being in state 2 and in group II decreases with the same possibilities as in the state 2 and group I case, but increases with ET from the centers of group I (migration). The same is true for the other center groups in state 2, but with its corresponding excitation sources. Equation 9 and the left-hand side level diagram of Figure 1 show that the microscopic probability of being in state 3 increases with the possibility of ET between the centers in state 2 of all groups and decreases with the possibility of intracenter decay. The u center net probability of being in state 2 at time t is

(5)

(2) (2) (2) ρ(2) u (t ) = ρu ,I (t ) + ρu ,II(t ) + ρu ,III(t ) + ...

NV

∑ M(rvu) ρ(0) v (t )

dρ(2) u (t )

v=1

dt

v=1

NV

∑ M(rvu) ρ(0) v (t ) v=1

NV

+2

∑ W (rvu) ρ(2) v (t )} v=1

v=1 NV

(11)

which is the correct differential equation for the migrationassisted up-conversion process.33,34,36 The Förster-like method45−51 of solving the donor probability average, ⟨ρD(est)(t)⟩, has been largely demonstrated to be the right procedure that retains the statistical ET expected effects. However, starting from a differential equation of the form of eq 11 turns this method impractical. Therefore, we must resort to another

∑ M(rvu) ρ(0) v (t ) v=1

NV

+2

v=1

NV

∑ M(rvu) ρ(2) v ,I (t )

(2) − ρ(2) u ,II(t ){γr−nr +

∑ M(rvu) ρ(2) v (t )

(2) − ρ(2) u (t ){γr−nr +

(6)

dt

NV

= ρ(0) u (t )

(2) (2) ∑ W (rvu)(ρ(2) v ,I (t ) + ρv ,II(t ) + ρv ,III(t ) + ...)}

= ρ(0) u (t )

(10)

We can check the validity of having divided state 2 into groups by adding the equations for state 2 (eqs 6, 7, 8, and so on) and applying eq 10:

NV

dρ(2) u ,II(t )

(2) (2) (3) (3) ∑ W (rvu)(ρ(2) v ,I (t ) + ρv ,II(t ) + ρv ,III(t ) + ...) − ρu (t )γr−nr

(9)

,where each ρu(est)(t) is the possibility or probability for u to be in the particular state at time t. As a quantum mechanical probability of being in a state, it is a continuous real number between 0 and 1. It will depend on the particular environment of interacting centers around u; therefore, to reach experimentally related averaged quantities, we have to submit the collections of interest, {ρu(2)(t)} and {ρu(3)(t)}, to some statistics. Differential equations for the temporary evolution of the microscopic probabilities of being in the luminescent states 2 and 3 and in groups I, II, III, ... at time t are

+2

(2) (2) = (ρ(2) u ,I (t ) + ρu ,II(t ) + ρu ,III(t ) + ...)

v=1

(2) (2) (2) ρ(0) u (t ) + ρu ,I (t ) + ρu ,II(t ) + ρu ,III(t ) + ...

dt

(8)

NV

×

Extending this relationship to each particular optical center u, we have the condition

(2) = −ρ(2) u ,I (t ){γr−nr +

(2) ∑ W (rvu)(ρ(2) v ,I (t ) + ρv ,II(t )

v=1 (2) + ρv ,III(t ) + ...)}

(4)

u = 1, 2, 3, ..., NV

NV

∑ M(rvu) ρ(0) v (t )

NV

(2) (2) (3) ρ̅(0)(t ) + ρ̅(2) I (t ) + ρ̅ II (t ) + ρ̅ III (t ) + ... + ρ̅ (t ) = 1

dρ(2) u ,I (t )

v=1 (2) − ρ(2) u ,III(t ){γr−nr +

where, for analysis purposes, the population that remains in the (2) (t), the group second state 2 at time t was divided into (i) NI,V of optical centers that were initially excited by the external (2) source, (ii) NII,V (t), the group of optical centers that reached (2) state 2 by ETM from the initially excited group, (iii) NIII,V (t), the group of optical centers that reached state 2 by ETM from (2) the group NII,V (t), and so on. Any other states different from the relevant ones are unimportant to the analysis, and their effective population is negligible in comparison to the population of the ground state 0 and to the population of state 2, which are the more populated states and those directly related to the relevant macroscopic parameters. Dividing by NV, we obtain a general condition for the mean probabilities of being in any relevant state, ρ(est) ̅ (t) (est = 0, 2, 3), and, in the case of state 2, of belonging, simultaneously, to a particular group, I, II, III, etc.:

+ ρ(3) u (t ) = 1

NV

∑ M(rvu) ρ(2) v ,II(t )

= ρ(0) u (t )

(2) (2) ∑ W (rvu)(ρ(2) v ,I (t ) + ρv ,II(t ) + ρv ,III(t ) + ...)} v=1

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starting point by returning to eqs 6−8. First, it should be noted that most of the time the importance order among the magnitudes of the populations is (2) (2) NI,(2) V (t ), NII, V (t ) > NIII, V (t ) > ...

dN (3)(t ) = NI(2)(t )[WII(t ) + WIII(t )] dt (2) (t )[WIII(t ) + WIIII(t )] − N (3)(t )γ(3) + NII r−nr

(12) (2) ρu,I (t),

(19)

(2) ρu,II (t),

Therefore, the first two probabilities in eq 10, are the most important to account for the ρu(2)(t) probability. To simplify the analysis, we can do the approximation of truncating the chain of differential equations for ρu(2)(t) and leave (2) (2) only the first two groups, NI,V (t) and NII,V (t), but taking the precaution to incorporate in group 2 the excitation of the other despised groups. This can be done also by truncating the (2) migration deactivation of the group NII,V (t). If this were not done, we would assume that once the migration takes place from group I to group II, the group II excitation is quenched by an additional process similar to a cross-relaxation mechanism, but with the rate of the migration process. We will see that these approaches are sufficient to generate behavior in accordance with the experimental trends. Applying the above considerations, the reformulated set of differential equations are dρ(2) u ,I (t )

(2) = −ρ(2) u ,I (t ){γr−nr +

dt

NV

∑

where MI(t ) =

=

WIII(t ) + WIIII(t ) =

NV

v=1 NV

(2) ∑ W (rvu)(ρ(2) v ,I (t ) + ρv ,II(t ))} v=1

(14)

dρ(3) u (t ) dt

(2) = (ρ(2) u ,I (t ) + ρu ,II(t ))

NV

∑ W (rvu)(ρ(2) v ,I (t )

V

=

−2

(15)

dt

(22)

NV

∑ W (rvu) ∫

0

t

t (0) ρ (t )dt 0 v

∑ M(rvu) ∫

(2) [ρ(2) v ,I (t ) + ρv ,II(t )]dt } (23)

NV(est)(t ) V

= N (est)(t ) =

N

∑

(2) ρu,I (0)

,where = 1 or 0 if the u center is or is not initially excited, respectively. As indicated and justified in previous papers,49−51 the influence of the integral functionalities in eq 23 on the final average, ρ̅I (2)(t), is low, so we can adopt some simple approximations:

ρ(est) u (t )

u=1

(2)

(2) −γr−nrt ρ(2) v ,I (t ) ≈ ρv ,I (0)e

= −NI(2)(t ){γ(2) r−nr + MI(t ) + 2[WII(t ) + WIII(t )]}

(2) dNII (t )

(2) NII, V (t ) u = 1 v = 1

v=1

,with V, the volume of the solid, permits transformation of eqs 13−15 into macroscopic equations:

dt

∑ ∑ W (rvu)

v=1

(16)

dNI(2)(t )

(21)

NV NV

1

NV

with u = 1, 2, 3, ..., NV. The relationship N

∑ ∑ W (rvu)

NI,(2) V (t ) u = 1 v = 1

(2) (2) ρ(2) u ,I (t ) = ρu ,I (0)exp{−γr−nrt −

v=1 (2) (3) + ρv ,II(t )) − ρu (t )γ(3) r−nr

∑u =V1 ρ(est) u (t )

NV NV

1

are macroscopic time-dependent ET rates. We will see that the strategy of expressing the migration flux in terms of NI(2)(t) MI(t) instead of N(0)(t) MI′(t) in eq 18 permits great simplification of the problem solution. Note that it could be possible because both interaction sums in eq 20 are the same. The partial solution of eq 13 is

∑ M(rvu) ρ(2) v ,I (t )

(2) − ρ(2) u ,II(t ){γr−nr + 2

(20)

(2) (2) × ρ(2) u ,II(t ){ρv ,I (t ) + ρv ,II(t )}

(13)

v=1

dt

(0) ∑ ∑ M(rvu) ρ(2) v ,I (t ) ρu (t )

M(rvu) ρ(0) v (t )

(2) ∑ W (rvu)(ρ(2) v ,I (t ) + ρv ,II(t ))}

= ρ(0) u (t )

1

NI,(2) V (t ) u = 1 v = 1

(2) (2) × ρ(2) u ,I (t ){ρv ,I (t ) + ρv ,II(t )}

NV

dρ(2) u ,II(t )

(0) ∑ ∑ M(rvu) ρ(2) u ,I (t ) ρv (t )

NI,(2) V (t ) u = 1 v = 1 NV NV

WII(t ) + WIII(t ) =

v=1

+2

NV NV

1

(2) ρ(0) v (t ) ≈ 1 − ρv ,I (t ) ≈ 1

(2) −γ(2) r−nr t ≈ 1 − ρ̅(2)(0)e−γr−nr t − ρ(2) v ,I (0)e I

(17)

(24)

The first one is justified in refs 49−51, and the second follows (2) from eq 5 with ρu,III (t) + ... ≈ 0 and the fact that the order of importance in the population in most experimental situations and at times near the origin is

= NI(2)(t ) MI(t ) (2) I II − NII (t ){γ(2) r−nr + 2[WII(t ) + WII (t )]}

(2) (3) NV(0)(t ) ≈ NV > NI,(2) V (t ) > NII, V (t ), NV (t )

(18) 7137

(25)



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(2) (2) Therefore, as ρν,I (t) > ρν,II (t), near the time origin, preliminarily we obtain

(2) (2) ρ(2) v ,I (t ) + ρv ,II(t ) ≈ ρv ,I (t )

We need a rough approximation of NI(2)(t) MI(t) consistent with eq 30. Also defining auxiliary time averages or effective rates in eq 17, we have

(26)

NI(2)(t ) MI(t ) = −

Soon we will lift this approximation, and then, in the Results and Discussion, we will check its validity. With those approximations, developing the integrals and rearranging, we have

⎧ (2) ⎫ ⎪ 1 − e−γr−nrt ⎪ ⎬ exp⎨−2W (rvu) (2) ⎪ ⎪ γ v=1 ⎭ ⎩ r−nr

× e−γII,eff t

(27)

where we have made = in the second sum and we have transformed the exponential of the sum into a product of exponentials. In ref 51 we indicated how to calculate (2) (t)⟩ = ρ̅I(2)(t) for expressions of mathematical the average ⟨ρu,I form similar to eq 27. It is based on a modified49−51 Förster-like procedure.45,46,49 The result for electric multipolar interaction is

−

∫0

t

MI(t )dt

(35)

We adopt a first-order approximation of NII(2)(t) (or of γI,eff ≈ γII,eff ≈ γ(2) r−nr

(36)

which improves as the optical densities or the excitation intensities decrease. This approximation means that in eq 18 we have additionally eliminated WIII(t) + WIIII(t) before solving the differential equation for NII(2). This ensures that NII(2) lasts a little longer, which helps to mitigate the failure to consider additional migration steps between the other groups. With this approximation eq 34 becomes

(2) 1 − e−γr−nrt (2) TM(t ) ≡ t − ρ̅ I (0) γ(2) r−nr

∫

(2) 1 − e−γr−nrt

t

(2) − MI(t )dt (2) NII (t ) ≈ NI(2)(0)e−γr−nrt {1 − e 0 }

(29)

(37)

where we have used eq 35. By using eq 37, we can lift the restriction of eq 26 to improve expression 29. First, we need a very simple approximation of MI(t). Rewriting eq 20, we obtain

(2) ⟨ρu,II (t)⟩

On the other hand, to get an approximation of = ρ̅I(2) I (t) (eq 14), we can start with its related equation for the population variation (eq 18) and make the auxiliary approximation of considering WIII(t) and WIIII(t) “time averages” or “time-independent effective rates” (for each collection of exI II and WII,eff . With these considerperimental parameters), WII,eff ations and taking into account that NII(2)(0) = 0, the partial solution of eq 18 is

MI(t ) =

1 N

NV

∑

∑u =V1 ρ(2) u ,I (t ) u = 1

NV

(0) ρ(2) u ,I (t ){ ∑ M(rvu) ρv (t )} v=1

NV

≡ ⟨ ∑ M(rvu) ρ(0) v (t )⟩N (2)(t ) I, V v=1

(30)

(38)

(2) We note that MI(t) is an average (among NI,V (t) optical centers) of a sum of microscopic time-dependent interactions. (2) (t) ≈ ρ̅I(2)(t) and ρν(0)(t) ≈ 1 By using the approximations ρu,I (eq 25), this average becomes a time-independent expected

where I II γII,eff ≡ γ(2) r−nr + 2[WII,eff + WII,eff ]

(34)

ρ̅I(2) I (t)):

where

t γ e II,eff t NI(2)(t ) MI(t )dt

t γ e II,eff t NI(2)(t )dt

NI(2)(t ) = NI(2)(0)e−γI,eff t e

(28)

∫0

∫0

where NI(2)(t) appropriate for these approximations can be found by solving the differential equation in eq 32:

(2) (2) ρ̅(2) I (t ) = ρ̅ I (0)exp[−γr−nrt ] ⎤ ⎡ 4π × exp⎢ − N Γ[1 − 3/s](ΩM TM(t ))3/ s ⎥ ⎦ ⎣ 3 ⎡ 4π (2) ⎤ × exp⎢ − N ρ̅ I (0)Γ[1 − 3/s](2ΩW TW (t ))3/ s ⎥ ⎣ 3 ⎦

(2) NII (t ) = e−γII,eff t

(33)

(2) NII (t ) = e−γII,eff t NI(2)(0) − NI(2)(t ) + [γII,eff − γI,eff ]

N v(2)(0) ∑v=1 1

γ(2) r−nr

(32)

By substituting expression 32 into expression 30, integrating by parts, and reordering

∏

TW (t ) ≡

− NI(2)(t )γI,eff

I II γI,eff ≡ γ(2) r−nr + 2[WI,eff + WI,eff ]

NV(2)(0)

Nv (2) ∑v=1 ρν,I (0)

dt

where

(2) (2) ρ(2) u ,I (t ) = ρu ,I (0)exp{−γr−nrt } ⎧ ⎡ (2) ⎤⎫ NV ⎪ 1 − e−γr−nrt ⎥⎪ ⎢ (2) ⎨ × ∏ exp −M(rvu)⎢t − ρ̅ I (0) ⎥⎬ (2) ⎪ γ ⎢ ⎥⎦⎪ v=1 ⎣ r−nr ⎩ ⎭

×

dNI(2)(t )

(31) 7138



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value of the microscopic rate sum for a random distribution, which is equivalent to the calculation of MI(0):49 MI(t ) ≈

NV

1 NV

This permits definition of the macroscopic time-dependent rates implied in eq 17:

NV

MI(t ) ≡

∑ { ∑ M(rvu) ρ(0) v (t )}

(2)

u=1 v=1 RV 4πr 2 ΩM , s =N dr rs Rm V

−γr−nrt × (1 − ρ̅(2) ) I (0)e

∫

=

4π N Γ[1 − 3/s](ΩM , s)3/ s (TM(t ))3/ s − 1 s

WII(t ) + WIII(t ) ≡

4πN ΩM , s s−3 (s − 3)R m

(39)

where Rm is the minimal approach distance between two ions in the lattice. With this expression and taking into account ρI̅ (2) I (t) = NII(2)(t)/N and ρI̅ (2)(0) = NI(2)(0)/N in eq 37, a simple expression (2) for ρν,II (t) is

(44)

4π (2) N (0)Γ[1 − 3/s](2ΩW , s)3/ s s V ⎛ ⎜ (2) × (TW (t ))3/ s − 1⎜2e−γr−nrt ⎜ ⎜ ⎝ ⎡ 4πN ΩM , s ⎤ ⎞ ⎥t ⎟ −⎢γ(2) + − r nr s−3 ⎥ (s − 3)R m ⎦⎟ − e ⎢⎣

s−3 (2) −γ(2) r−nrt {1 − e−4πN ΩM , s /(s − 3)R m t } ρ(2) v ,II(t ) ≈ ρv ,I (0)e

⎟ ⎟ ⎠

(40)

Equation 44 could be substituted into eq 37 to obtain

which completes the approximation of eq 24. Therefore, in eq 13

(2)

(2) NII = NI(2)(0)e−γr−nrt

⎧ ⎡ 4πN ΩM , s ⎤ ⎫ ⎥t ⎪ ⎪ −⎢γ(2) (2) r−nr + s−3 ⎥ ⎪ (2) (2) (2) ⎪ ⎢ −γ t (s − 3)R m ⎦⎬ ρv ,I (t ) + ρv ,II(t ) ≈ ρv ,I (0)⎨2e r−nr − e ⎣ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

3/ s × {1 − e−(4π /3N )Γ[1 − 3/ s](ΩM , sTM(t )) }

(41)

(2) N (2)(t ) = NI(2)(t ) + NII (t )

⎧ ⎡ 4πN ΩM , s ⎤ ⎫ ⎥t ⎪ −⎢γ(2) + t⎪ ⎪ −γ(2) t − r nr s−3 ⎥ ⎪ (s − 3)R m ⎦ ⎬ dt ⎨2e r−nr − e ⎢⎣ TW (t ) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

(2) = N ρ̅(2) I (0)exp[−γr−nrt ]

⎤ ⎡ 4π × exp⎢ − N Γ[1 − 3/s](ΩM TM(t ))3/ s ⎥ ⎦ ⎣ 3

∫

⎡ 4π × exp⎢ − N ρ̅(2) I (0)Γ[1 − 3/s] ⎣ 3

(2) 2 − 2e−γr−nrt = γ(2) r−nr

(2) ⎤ × (2ΩW TW (t ))3/ s ⎥ + N ρ̅(2) (0)e−γr−nrt I ⎦

⎛ ⎡ 4πN ΩM , s ⎤ ⎞ ⎥t −⎢γ(2) + ⎜ − r nr s−3 ⎦ ⎟ s − 3⎜1 − e ⎣ (s − 3)R m ⎟ (s − 3)R m ⎜⎜ ⎟⎟ ⎝ ⎠

3/ s × {1 − e−(4π /3N )Γ[1 − 3/ s](ΩM TM(t )) }

(47)

On the other hand, for the up-conversion state, taking into account that N(3)(0) = 0, the partial solution of eq 19 is

s−3 4πN ΩM , s + γ(2) r−nr(s − 3)R m

(3) N (3)(t ) = e−γr−nrt

(42)

With this function and as NI(2)(t) = Nρ̅I(2)(t), the derivative of expression 28 with respect to time gives an explicit form of the differential eq 17: dNI(2)(t ) dt

(46)

This expression added to eq 28 (times N) completes the total population that remains in state 2 at time t:

which permits TW(t) of eq 29 to be redefined:

−

(45)

∫0

t γ(3) t (2) e r−nr NI (t )[WII(t ) + WIII(t )]

(3) × dt + e−γr−nrt

⎧ ⎪ ⎪ 4π 3/ s = −NI(2)(t )⎨γ(2) r−nr + s N Γ[1 − 3/s](ΩM , s) ⎪ ⎪ ⎩

∫0

t γ(3) t (2) e r−nr NII (t )

× [WIII(t ) + WIIII(t )]dt

(48)

Similarly to the MI(t) case in eqs 38 and 39, we can rewrite eqs 21 and 22

(2)

−γr−nrt ) × (TM(t ))3/ s − 1(1 − ρ̅(2) I (0)e 4π (2) N (0)Γ[1 − 3/s](2ΩW , s)3/ s (TW (t ))3/ s − 1 + s V ⎛ ⎡ 4πN ΩM , s ⎤ ⎞⎫ ⎥t ⎟⎪ ⎜ −⎢γ(2) (2) t r−nr + s−3 ⎥ ⎪ ⎢ −γ (s − 3)R m ⎦ ⎟⎬ × ⎜2e r−nr − e ⎣ ⎜ ⎟⎪ ⎜ ⎟⎪ ⎝ ⎠⎭ (43)

WII(t ) + WIII(t ) =

1

NV

ρ(2) u ,I (t ) NV (2) ∑u = 1 ρu ,I (t ) u = 1 NV (2) × [ ∑ W (rvu){ρ(2) v ,I (t ) + ρv ,II(t )}] v=1

∑

(49) 7139


The Journal of Physical Chemistry C WIII(t ) + WIIII(t ) =

NV

1

∑

N

∑u =V1 ρ(2) u ,II(t ) u = 1

Article

by migration is added into the temporal function, TW(t) (eq 42, NI(2), full curve), only a small additional deviation appears with (2) respect to the case where it is not considered (NI,approx , eq 29, dashed curve), which confirms that the integral functionality (eqs 23, 28, 29, and 42) influence is small. For simplicity, we will use the TW(t) expression of eq 29 for the rest of the graphical analysis. The rising−falling curves of Figure 3 show the optical density time variation for the group that reaches state 2 by energy

ρ(2) u ,II(t )

NV

(2) × [ ∑ W (rvu){ρ(2) v ,I (t ) + ρv ,II(t )}] v=1 (50)

and use the approximations ρ̅I(2) I (t) to get

(2) ρu,I (t)

≈

ρ̅I(2)(t)

and

(2) ρu,II (t)

NV NV 1 I II WI (t ) + WI (t ) = ∑ [ ∑ W (rvu) NV u=1 v=1 (2) × {ρ(2) v ,I (t ) + ρv ,II(t )}] = WIII(t ) + WIIII(t )

≈

(51)

Therefore, expression 48 becomes (3) N (3)(t ) = e−γr−nrt

∫0

t γ(3) t (2) e r−nr N (t )[WII(t ) + WIII(t )]dt

Figure 3. Time variation for the state 2 population of groups I and II at two different concentrations, N1 = 3.0 × 1019 cm−3 (dashed curves) and N2 = 7.0 × 1019 cm−3 (full curves). ρ̅I(2)(0) = 0.2, ΩW,6 = 8.86 × (2) = 2.0 ms−1, and ΩM,6 = 8.86 × 106 Å6 ms−1 (RW,6 = 12.8 Å), γr−nr 106 Å6 ms−1 (RM,6 = 12.8 Å). The descendant curves correspond to NI(2)(t) of eqs 28 and 29 and the ascendant−descendant ones to NII(2)(t) of eq 46.

(52)

where N(2)(t) is given by eq 47 and WII(t) + WIII(t) by eq 45.

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RESULTS AND DISCUSSION Figure 2 shows semilogarithmic plots for the state 2 optical density time variation of the group that was initially excited,

migration, NII(2)(t), eq 46, at two different concentrations, N1 = 3.0 × 1019 cm−3 (dashed curve) and N2 = 7.0 × 1019 cm−3 (full curve), for dipole−dipole interaction and some typical parameters. As can be seen the trend is as expected: the ETM is a quick event that lasts some time and increases with the dopant optical concentration at the same excitation power (constant ρ̅ I(2)(0) = 0.2 value). We can compare this behavior with the initially excited population, NI(2)(t), eq 28, which decreases with time as shown in the figure: at the smaller concentration, N1 = 3.0 × 1019 cm−3 (dashed curves), and for the parameters shown in the figure caption, the NII(2)(t) population belatedly reaches NI(2)(t). When the concentration is higher, N2 = 7.0 × 1019 cm−3 (full curves), the NII(2)(t) population quickly exceeds NI(2)(t) because, being more probable, there are more migration events, which results in an increase in NII(2)(t) and a faster decrease of NI(2)(t). To better appreciate the concentration effect on intensive properties, the inset in Figure 3 shows the concentration effect on the probability of being in state 2, ρ̅(2)(t) = N(2)(t)/N(2)(0) for each group I and II. As can be seen, the curvatures of ρ̅I(2)(t) and ρ̅I(2) I (t) are very sensitive to the optical density increase, which is a reasonable result, because both up-conversion and migration effective macroscopic ET rates are concentration dependent. As can be seen in Figure 4 a similar behavior compared to the previous case is observed by varying the external excitation power (through ρ̅I(2)(0)) for some typical parameters (see the figure caption) and dipole−dipole interaction. However, as can be appreciated in the figure inset, the ρ̅I(2)(0) and ρ̅I(2) I (t) probabilities are less sensitive to changes in ρ̅I(2)(0). Figure 5 shows that the curvatures of NI(2)(t) and NII(2)(t) are very sensitive to changes in migration intensity constants: ΩM,6 = 1.0 × 106 Å6 ms−1 (RM,6 = 8.9 Å) for the short-dashed curves and ΩM,6 = 4.0 × 107 Å6 ms−1 (RM,6 = 16.5 Å) for the full curves, keeping the other parameters constant (see the figure

Figure 2. Semilogarithmic plots for the temporal variation of the initially excited state 2 population (eq 28 times N). N = 5.0 × 1019 cm−3, ρ̅I (2)(0) = 0.2, ΩW,6 = 8.86 × 106 Å6 ms−1 (RW,6 = 12.8 Å), Rm = (2) (2) 3.5 Å, and γr−nr = 2.0 ms−1. The dotted curve, NI,M , is for the pure up0 conversion case (without migration, ΩM,6 = 0.0 Å6 ms−1). The full curve, NI(2), was obtained using expression 42 and the dashed curve, (2) , using expression 29 for TW(t). In both cases ΩM,6 = 8.86 × NI,approx 106 Å6 ms−1(RM,6 = 12.8 Å).

NI(2)(t), for some typical parameters and dipole−dipole interaction. By typical parameters we mean values usually found in Ln3+-doped amorphous solids, including the critical transfer distances RM and RW (ΩW,s and ΩM,s terms).19,21,33,48 Equation 28 is plotted using expressions 42 (NI(2), full curve) (2) , dashed curve) and is compared with the case and 29 (NI,approx (2) where the migration is absent (NI,M , pure up-conversion, dotted 0 curve) for the same set of parameters. When the migration process is considered, a more pronounced deviation from simple exponential decay loss is obtained. This is an expected trend, because the migration possibility increases the statistical deactivation of the initially excited population even more. Also, we can observe that when the effect of the population excited 7140



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Figure 6. Comparison, on a semilogarithmic scale, among N(2)(t) values for the simple exponential decay case (“Exp”, full straight line, (2) equation NI (2)(0)e−γr−nrt), the pure up-conversion case (ΩM,6 = 0.0 Å6 ms−1, N (2) I,M0 , dotted curve, eq 28), and the migration-assisted upconversion case, eq 28, with ΩM,6 = 7.0 × 106 Å6 ms−1 (RM,6 = 12.3 Å, (2) N I,M , dashed curve) and ΩW,6 = 4.0 × 107 Å6 ms−1 (RM,6 = 16.5 Å, 1 (2) (2) N I,M2 , long-dashed curve). N = 5.0 × 1019 cm−3, ρ u,I (t) ≈ ρ(2) I (0) = 0.2, 6 6 −1 ΩW,6 = 8.86 × 10 Å ms (RW,6 = 12.8 Å), andr−nr γ (2) = 2.0 ms−1.

Figure 4. Time variation for the state 2 probabilities of groups I and II at two different external excitation powers (through ρ̅I (2)(0)), ρ1 = (ρ̅I(2)(0) =) 3.0 × 1019 cm−3 (dashed curves) and ρ2 = (ρ̅I(2)(0) =) 7.0 × 1019 cm−3 (full curves). N = 5.0 × 1019 cm−3,ΩW,6 = 8.86 × 106 Å6 ms−1 (2) (RW,6 = 12.8 Å), γr−nr = 2.0 ms−1, and ΩM,6 = 8.86 × 106 Å6 ms−1 (RM,6 = 12.8 Å). The descendant curves correspond to NI(2)(t) of eqs 28 and 29 and the ascendant−descendant ones to NII(2)(t) of eq 46.

(2) ΩM,6 = 4.0 × 107 Å6 ms−1 (N I,M , long-dashed curve) for the 1 same values of the other parameters. The initial slope of the curves indicates that the pure up-conversion and the migrationassisted cases start deactivating with the same rate, but soon they differ: the population affected solely by the up-conversion interaction is quenched faster than the cases where the migration is present, and the deactivation is slower and closer to the simple exponential decay as the migration intensity constant increases. As we see, our model actually agrees with the known experimental observation that migration increases the residence time of the quenched population to favor the upconversion process. We can also observe that the model accounts for the statistical effects of deviation from simple exponential decay observed in ET processes in solids. Figure 7 shows curves for the up-converted population time variation (eqs 45, 47, and 52) for two different concentrations

Figure 5. Time variation for the state 2 populations at two different effective migration constants: ΩM,6 = 1.0 × 106 Å6 ms−1 (RM,6 = 8.9 Å) for the short-dashed curves and dotted curve and ΩM,6 = 4.0 × 107 Å6 ms−1 (RM,6 = 16.5 Å) for the full curves and long-dashed curve. N = 5.0 × 1019 cm−3, ρI (2)(0) = 0.2, ΩW,6 = 8.86 × 106 Å6 ms−1 (RW,6 = 12.8 Å), (2) = 2.0 ms−1. The rising−falling curves correspond to the and γr−nr (2) NII (t) group (eq 46). The first two falling internal curves correspond to NI (2)(t) of eqs 28 and 29, and the last two falling external curves correspond to the total population, N(2)(t) = NI(2)(t) + NII(2)(t) (eq 47).

caption). Notice how, as the migration intensity constant increases, the NII(2)(t) population also increases while the NI(2)(t) population decreases, as it should be. To see how this migration intensity constant change affects the total state 2 population (eq 47), we show this quantity in the dotted (ΩM,6 = 1.0 × 106 Å6 ms−1, RM,6 = 8.9 Å) and large-dashed (ΩM,6 = 4.0 × 107 Å6 ms−1,RM,6 = 16.5 Å) curves. We note that although the component groups NI(2)(t) and NII(2)(t) are very sensitive to (2) changes in ΩM,6, the state 2 total population, ρ̅I(2) I (t) = N (t) = (2) (2) (2) NI (t) + NII (t), is not, because when the NII (t) component increases, the NI(2)(t) component decreases, canceling much of the effect. However, we can observe that the net effect of increasing the migration intensity is the increase in the effective state 2 lifetime, as expected. To better appreciate the migration intensity constant influence on the state 2 population and the statistical effects of the ET processes, in Figure 6 we plot, on a semilogarithmic scale, a comparison among N(2)(t) values in the simple exponential (2) decay case (“Exp”, full straight line, equation NI(2)(0)e−γ r−nrt), the pure up-conversion case (ΩM,6 = 0.0 Å6 ms−1, N (2) I,M0 , dotted curve, eq 28), and the migration-assisted up-conversion case, eq (2) , dashed curve) and 28, with ΩM,6 = 7.0 × 106 Å6 ms−1 (N I,M 1

Figure 7. Time variation of the up-converted population (eqs 45, 47, and 52). The first three curves from the bottom are for the optical density of N1 = 5.0 × 1019 cm−3 with ΩM,6 = 0.0 Å6 ms−1 (NM(3)0N1 , dotted curve, pure up-conversion case), ΩM,6 = 8.86 × 106 Å6 ms−1 (RM,6 = 12.8 Å,NM(3)1N1 , dashed curve), and ΩM,6 = 8.86 × 106 Å6 ms−1 (RM,6 = 27.6 Å,NM(3)2N1 , full curve). The three higher curves are for the optical density of N2 = 7.0 × 1019 cm−3 with ΩM,6 = 0.0 Å6 ms−1 (NM(3)0N2 , dotted curve, pure up-conversion case), ΩM,6 = 8.86 × 106 Å6 ms−1 (RM,6 = 12.8 Å, NM(3)1N2 , dashed curve), and ΩM,6 = 8.86 × 108 Å6 ms−1 (RM,6 = 27.6 Å, NM(3)2N2 , full curve). ρI(2)(0) = 0.2, ΩW = 8.86 × 106 (2) (3) Å6 ms−1 (RW,6 = 12.8 Å), γr−nr = 2.0 ms−1, and γr−nr = 3.0 ms−1.

and, for each concentration, three different migration intensity constants. The three smaller curves correspond to the optical 7141



■

density N1 = 5.0 × 1019 cm−3 and the three higher ones to N2 = 7.0 × 1019 cm−3. As we can see a slight increase in optical density, N, greatly favors the up-converted population as should be, because in the case analyzed here the up-converted population should roughly vary as the optical density to the second power.15,37,38,42,50 The populations are less sensitive to changes in migration intensity constants, as can be observed for the group of curves for each concentration, but in general, as the migration intensity constants increase, the up-converted population increases too. In this respect we can appreciate how this model is consistent with the known experimental results of up-converted population increasing when energy migration is present (compare the pure up-conversion cases, dotted curves, with the migration-assisted cases, dashed and full curves). Finally, eqs 28, 46, 47, and 52 can be used in the fitting of experimental transient curves. For example, if we want to determine the critical transfer distances RM and RW (ΩW,s and ΩM,s terms, eq 2) and the interaction multipoles, we should first estimate the used optical center concentration, N, the initially excited optical center fraction, ρ̅ (2) I (0), the state 2 lifetime in the −1 absence of ET, (γ (2) r−nr) , and the minimal approach distance between two ions in the lattice, Rm. All of them can be estimated independently from the transient curves under analysis. As we have two critical transfer distances, RM and RW, to determine, we recommend first finding RM in a very low concentration sample, where the ETU is negligible. Therefore, we can use eq 46 solely in the first fitting and use the found RM value in the second fitting at a greater concentration where the ETU is present. Of course, and as usual,48 a fitting should be obtained for each multipolar interaction value supposed. The best fit corresponds to the best value of the sought parameters.

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], czhadad@ gmail.com. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS F.F. acknowledges Grant UNAB-DI-01-11/I for partial funding of this work and also thanks Conicyt for Graduate Fellowship Number 63100014. C.Z.H. acknowledges financial support from the Universidad de Antioquia.

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REFERENCES

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CONCLUSIONS The plots show that the model is physically consistent and correctly reproduces the literature experimental trends: (i) a lifetime luminescence increase of the state from which the energy migrates with respect to the case without migration and, concomitantly, an increase of the up-converted luminescence as the migration rate increases, (ii) the known simple exponential decay loss of the quenched state luminescence, typical of the ET processes in solids. This model predicts that this last effect is more pronounced at the beginning and tends to the simpleexponential decay at longer times when the ET events are less frequent. We conclude that this model could be used in the fitting of experimental curves to obtain information about the different parameters involved in the mixed ET processes. Additionally, the present study has shown that (i) it is possible to build a migration-assisted up-conversion model that retains the successful and widely proven Förster-like method to calculate the quenched state averages, (ii) through one-step migration that incorporates a luminescence retention time it is possible to correctly account for the experimental trends of the migration-assisted up-conversion phenomenon, and (iii) by means of a model that artificially separates the initially excited population, N (2) I (t), from the migration excited population, (t), it is possible to reach some understanding about the N(2) II factors that affect the observable population variation and to study the behavior of the components separately and their interrelation. For example, it was found that despite the fact (2) that both components N(2) I (t) and N II (t) are very sensitive to changes in concentration and in ΩM,6, N(2)(t) is sensitive only to a lesser degree. 7142



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Up-Conversion and Migration by Energy ... - American Chemical Society

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