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On the luminescence quenching of F centres in alkali halides
Solid State Communications, Vol. 33, pp. 907-910. Pergamon Press Ltd. 1980. Printed in Great Britain. ON THE LUMINESCENCE QUENCHING OF F CENTRES IN ALKALI HALIDES C.H. Leung and K.S. Song Dept. of Physics, University of Ottawa, Ottawa, Canada (Received 8 November 1979 by R. Barrie)
Non-radiative transitions from the excited state of F center to the ground state occur very efficiently around and above, over a rather large interval of energy, the crossing point of the two adiabatic potential energy curves. Taking this effect into account the criterion for luminescence quenching originally proposed by Dexter et al. is re-examined. The luminescence quantum efficiency is estimated for those alkali halides in which the luminescence is weak, and the results are in general accord with experiment. The border-line case of NaC1 is also explained. THE PRINCIPAL FEATURES of light absorption and subsequent luminescence by F centers in alkali halides are well understood. A model often used to explain the phenomena consists of two electronic states coupled to the lattice linearly through an effective mode of vibration. The absence of luminescence in some of the alkali halides was first studied by Dexter et al. (DKR) [1] who attributed this to the fact that the excited system undergoes a non-radiative transition to the ground state before reaching the new equilibrium point on the excited state energy surface. This qualitative proposition has recently been reformulated more explicitly by Bartram and Stoneham [2]. They introduce a parameter A which indicates whether the excited state optically prepared is above (A > ¼) or below (A < ¼) the crossing point of the excited and ground state adiabatic potential energy surfaces. They also studied the competition at the crossing point between two processes: remaining on the excited state and making a non-radiative transition to the ground state [3]. In a recent work on the relaxation of the excited self-trapped exciton in alkali halides [4], we have found that the non-radiative transition (NRT) is very efficient over a rather large energy interval above and around the crossing point. This suggests that a further refinement of the criterion for luminescence quenching is desirable. In this short note, we present a model which takes into account NRT above the crossing point as well as the effect of the damping of the lattice vibration. We have also estimated the quantum efficiency for luminescence of F centers employing a naive, but reasonable argument. The results are in qualitative agreement with experimental data. In the simplest model, the electron states of a F center are assumed to couple linearly to a single
effective mode Q+ft of frequency Weft. The configuration-coordinate diagram (Fig. 1) shows the total potential energy as a function of Qeff for the ground state Lg)and the first excited state Ix), as well as the optical absorption (A ~ B) and emission (C-* D) transitions. Vibronic wavefunctions are denoted as Ig)lm, g) and Ix)In. x), where m (or n) specifies the vibrational level of the Qeft mode in the state Ig)(or Ix)). In what follows we define various useful parameters with reference to Fig. I. First the Huang-Rhys factor is given by EBc S - h 6Oef t'
(1)
where Eric denotes the difference in energy between the points B and C in Fig. 1. Next we define an integer p by Ec~t P ~ h Wen"
(2)
In terms of S and p, the vibrational level n c of the excited state at the crossing point of the two potential curves is given by (P _ S) 2 nc .~ _ _ 4S
(3)
Finally, we include the parameter A, introduced by Bartram and Stoneham [2] :
A -
907
Shweft Eaa
(4)
908
ON THE LUMINESCENCE QUENCHING OF F CENTRES IN ALKALI HALIDES Vol. 33, No. 8
Table 1. Calculated luminescence efficiency ~ f o r various alkali halides. The parameters p, S and A are defined
Host
p
S
nc
A
rl
LiI NaI NaBr LiC1 LiBr LiF NaCI KI KBr RbI KF KC1 RbC1 NaF RbF RbBr
24 71 92 121 99 86 120 180 177 149 137 155 167 80 125 129
120 44 55 71 55 41 42 54 51 40 32 36 37 17 26 25
20 4 6 9 9 12 36 74 79 74 86 99 114 59 93 109
0.831 0.384 0.375 0.371 0.358 0.323 0.260 0.231 0.223 0.211 0.189 0.188 0.182 0.175 0.173 0.162
0.348 0.006 0.024 0.065 0.051 0.071 0.608
--~ 1.0
Using the values of S, N-Oen and Ena obtained from experiment [5], the values of p, n c and A for various alkali halides have been derived and listed in order of decreasing A in Table 1. These values will be useful later in our discussion• We discuss now the cooling process after the system is excited by optical absorption to the state [x)lno x). We confine ourselves only to the case of zero temperature• Consider the situation when the system is in the state Ix)In, x). The system may relax via two competing non-radiative processes: (i) intralevel transition (pure vibrational relaxation) to the next lower vibrational level Ix)In - 1, x ) with probability (per unit time) Vn, assuming multiphone processes are negligible, and (ii) inter-level transition to the vibronic state Ig)ln + p, g) of the same energy with probability W~. The first process describes the damping of the lattice vibration while the second is due to the coupling between Ig) and Ix) via the promoting mode. The fraction of F centers reaching the state Ix)In - 1, x ) is therefore V~/(V~ + W,). These F centers in the state Ix)In - I, x) will again relax down to Ix)In - 2, x ) with probability Vn-1 or tunnel to Ig)ln - 1 + p, g) with probability Wn-l , and the process continues. Thus starting with the state Ix)lno, x), the fraction o f F centers which reach the state Ix)J0, x) and give rise to luminescence is given by
,7=~f n=l
v. v. + w,,
(5)
Energy
I
__t_ 8
A
....
L
Fig. 1. Configuration-coordinate diagram to illustrate the absorption and emission processes for F centers• We remark that we have neglected here the effect of the back transitions from the ground state to the excited state, e.g. Ig)ln + p, g) ~ Ix)In, x), also of probability Wn. We expect however that because n + p is in general very large (of the order of a hundred, see Table 1), the probability of intra-level transitions to lower vibrational levels is much larger than Wn when the system is on the ground state energy surface [3]. This point has also been discussed by Dexter and Fowler [6]. Thus equation (5) should be a good approximation. We now discuss the two probabilities Vn and W,,. According to Stoneham and Bartram [3] Vn = nTrA~
(6)
where Aco is a measure of the average coupling between Qeff and the lattice. Unfortunately, the value of Aco is difficult to obtain theoretically. Nevertheless a rough estimate of Aw may be made, since for most common defects Vn is of the order 101°-1012 sec -l [7]. For inter-level NRTs, we follow the expression of Pooley [8] which is an extension of the work of Huang and Rhys [9]. Confining to the case of zero temperature we obtain Wn = Wol(n, x l n + p,g)j2
(7)
where Wo is a physical factor involving the transition matrix element of the promoting mode between Ig) and Ix). The second factor is the square of the overlap integral of the two displaced harmonic oscillator wavefunctions (the Franck-Condon factor) which can readily be expressed in terms of the Laguerre polunomial L~ (S) [4, 10]. We have calculated Wn for various alkali halides.
Vol. 33, No. 8
ON THE LUMINESCENCE QUENCHING OF F CENTERS IN ALKALI HALIDES
For convenience the physical factor Wo is taken as 1013 sec-I [4, 11 ]. As expected our calculation shows that Wn is largest when n ~ n c , i.e. when the vibrational levels are close to the crossing point of the two potential curves. The largest value is about 2 x 1011 sec-i for all cases. As n decreases from n c , Wn decreases quite rapidly; at about 20 levels below n o Wn decreases by more than five orders of magnitude. For n >~ n c, I¢n oscillates within the range 1011-1012 sec-1 for at least some thirty to forty vibrational levels apart from some fluctuations (see Fig. 2 of [4] ). The important point here is that the probabilities of inter-level transitions from levels above the crossing point may be quite comparable to those from levels near the crossing point. We are now in a position to give a rough estimate of 7/. We consider first those alkali halides in Table 1 for which A ( ¼. For these alkali halides S ~ n c - 20 (see Table 1) which means that the level reached by optical absorption is at least some twenty levels below the crossing point. According to our calculation discussed above, the inter-level transition probability Wn is less than 10 7 sec -1 , which is much smaller than Vn(10 n ~ 1012 sec-1). Thus the quantum efficiency of luminescence 77 ~ 1. On the other hand for those alkali halides with A ) ¼, excluding NaC1 for the moment, the level reached by optical absorption is at least some thirty levels above the crossing point (S ~ n c + 30). Since Wn is quite comparable with V, for levels near or above the crossing point, we expect 7? to be quite small. As a rough estimate, we may assume Wn to be about 1011 sec -1 for twenty levels above or near n c and zero otherwise. Taking Vn as 1011-1012sec -1, we obtain from equation (5) 0 ~< 7? <~ 0.15. Thus for these alkali halides, luminescence is expected to have a low efficiency or even be completely quenched. Although our prediction is in general accora with that based on the DKR model, the present model in fact gives a more stringent criterion for the quenching of luminescence than that of the DKR model. For luminescence to be (almost) quenched, the present model requires not only S ) nc(i.e. A ) ¼) but that S be considerably larger than n c (e.g. S >~ n c + 30). Likewise for luminescence efficiency to be close to 1, S has to be considerably smaller than n c (e.g. S <~ n c + 20), as compared with S ( n c (i.e. A (¼) in the DKR model. In the case when S -~ n c , as in NaC1, the efficiency of luminescence can only be determined by detailed calculations based on equations (5)--(7). As a comparative study, we have calculated 7? for the alkali halides in Table 1 with A ) ¼ using equations (5)-(7). We have taken rrA6o to be 5 x 101o sec-1, and for convenience only up to forty
909
levels (n = 1 ~ 40) have been included in equation (5) in all cases. The results of this calculation are also presented in Table 1. The efficiency r/is only about a few % for these alkali halides except for LiI and NaCI, whose values of n c are relatively large. Luminescence of the F center in NaC1 has been observed with a quantum efficiency of 0.33 [12]. Bartram and Stoneham [3] suggested that luminescence in NaC1 is expected because 26% of the F band absorption can populate excited states with energy less than that of the crossing point. The present model provides an alternative explanation that because of the competition between inter-level and intra-level transitions as the system relaxes, luminescence efficiency may still be large ifS is larger than but close to n c . As for the case LiI, luminescence has not been observed. However, LiI is quite anomalous because the potential minimum of the excited state lies outside the potential well of the ground state (see Fig. 2 of [2] ). Thus even if r/is not small, luminescence may be quite difficult to observe. The present work may be considered as complementing the recent formulation of Bartram and Stoneham of the earlier idea of Dexter et al. In a very recent work, Kusunoki [13] has presented a quantitative theory of luminescence quenching which also takes explicit account of the coupling between Ix) and Ig) as well as the damping of the lattice vibration. Using the Landau-Zener theory of molecular collision, the author arrives at similar conclusion as we do here regarding the partial quenching of luminescence for cases with A >~ 0.25. A c k n o w l e d g e m e n t - This work was supported in part
by a grant from the National Science and Engineering Research Council of Canada. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
D.L. Dexter, C.C. Klick & G.A. Russel, Phys. Rev. 100,603 (1955). R.H. Bartram & A.M. Stoneham, Solid State C o m m u n . 17, 1593 (1975) A.M. Stoneham & R.H. Bartram, Solid S t a t e Electronics 21, 1325 (1978). K.S. Song& C.H. Leung, Solid State C o m m u n . 32, 565 (1979). R.K. Dawson& D. Pooley, Phys. Status Solidi 35, 95 (1969). D.L. Dexter & W.B. Fowler, J. Chem. Phys. 47, 1379 (1967). A.M. Stoneham, Theory o f Defects in Solids, p. 548. Oxford University Press (1975). D. Pooley, Proc. Phys. Soc. L o n d o n 87, 245 (1966). K. ttuang & A. Rhys, Proc. R . Soc. A 2 0 4 , 406 (1950).
910 10. 11.
ON THE LUMINESCENCE QUENCHING OF F CENTERS IN ALKALI HALIDES T.H. Keil, Phys. Rev. A140, 601 (1965). Y. Suzuki, M. Okumura & M. Hirai, J. Phys. Soc. Japan 47, 184 (1979).
12. 13.
Vol. 33, No. 8
S. Honda & M. Tomura, J. Phys, Soc. Japan 33, 1003 (1972). M. Kusunoki, Phys. Rev. B20, 2512 (1979).
Non-radiative transitions from the excited state of F center to the ground state occur very efficiently around and above, over a rather large interval of energy, the crossing point of the two adiabatic potential energy curves. Taking this effect into account the criterion for luminescence quenching originally proposed by Dexter et al. is re-examined. The luminescence quantum efficiency is estimated for those alkali halides in which the luminescence is weak, and the results are in general accord with experiment. The border-line case of NaC1 is also explained. THE PRINCIPAL FEATURES of light absorption and subsequent luminescence by F centers in alkali halides are well understood. A model often used to explain the phenomena consists of two electronic states coupled to the lattice linearly through an effective mode of vibration. The absence of luminescence in some of the alkali halides was first studied by Dexter et al. (DKR) [1] who attributed this to the fact that the excited system undergoes a non-radiative transition to the ground state before reaching the new equilibrium point on the excited state energy surface. This qualitative proposition has recently been reformulated more explicitly by Bartram and Stoneham [2]. They introduce a parameter A which indicates whether the excited state optically prepared is above (A > ¼) or below (A < ¼) the crossing point of the excited and ground state adiabatic potential energy surfaces. They also studied the competition at the crossing point between two processes: remaining on the excited state and making a non-radiative transition to the ground state [3]. In a recent work on the relaxation of the excited self-trapped exciton in alkali halides [4], we have found that the non-radiative transition (NRT) is very efficient over a rather large energy interval above and around the crossing point. This suggests that a further refinement of the criterion for luminescence quenching is desirable. In this short note, we present a model which takes into account NRT above the crossing point as well as the effect of the damping of the lattice vibration. We have also estimated the quantum efficiency for luminescence of F centers employing a naive, but reasonable argument. The results are in qualitative agreement with experimental data. In the simplest model, the electron states of a F center are assumed to couple linearly to a single
effective mode Q+ft of frequency Weft. The configuration-coordinate diagram (Fig. 1) shows the total potential energy as a function of Qeff for the ground state Lg)and the first excited state Ix), as well as the optical absorption (A ~ B) and emission (C-* D) transitions. Vibronic wavefunctions are denoted as Ig)lm, g) and Ix)In. x), where m (or n) specifies the vibrational level of the Qeft mode in the state Ig)(or Ix)). In what follows we define various useful parameters with reference to Fig. I. First the Huang-Rhys factor is given by EBc S - h 6Oef t'
(1)
where Eric denotes the difference in energy between the points B and C in Fig. 1. Next we define an integer p by Ec~t P ~ h Wen"
(2)
In terms of S and p, the vibrational level n c of the excited state at the crossing point of the two potential curves is given by (P _ S) 2 nc .~ _ _ 4S
(3)
Finally, we include the parameter A, introduced by Bartram and Stoneham [2] :
A -
907
Shweft Eaa
(4)
908
ON THE LUMINESCENCE QUENCHING OF F CENTRES IN ALKALI HALIDES Vol. 33, No. 8
Table 1. Calculated luminescence efficiency ~ f o r various alkali halides. The parameters p, S and A are defined
Host
p
S
nc
A
rl
LiI NaI NaBr LiC1 LiBr LiF NaCI KI KBr RbI KF KC1 RbC1 NaF RbF RbBr
24 71 92 121 99 86 120 180 177 149 137 155 167 80 125 129
120 44 55 71 55 41 42 54 51 40 32 36 37 17 26 25
20 4 6 9 9 12 36 74 79 74 86 99 114 59 93 109
0.831 0.384 0.375 0.371 0.358 0.323 0.260 0.231 0.223 0.211 0.189 0.188 0.182 0.175 0.173 0.162
0.348 0.006 0.024 0.065 0.051 0.071 0.608
--~ 1.0
Using the values of S, N-Oen and Ena obtained from experiment [5], the values of p, n c and A for various alkali halides have been derived and listed in order of decreasing A in Table 1. These values will be useful later in our discussion• We discuss now the cooling process after the system is excited by optical absorption to the state [x)lno x). We confine ourselves only to the case of zero temperature• Consider the situation when the system is in the state Ix)In, x). The system may relax via two competing non-radiative processes: (i) intralevel transition (pure vibrational relaxation) to the next lower vibrational level Ix)In - 1, x ) with probability (per unit time) Vn, assuming multiphone processes are negligible, and (ii) inter-level transition to the vibronic state Ig)ln + p, g) of the same energy with probability W~. The first process describes the damping of the lattice vibration while the second is due to the coupling between Ig) and Ix) via the promoting mode. The fraction of F centers reaching the state Ix)In - 1, x ) is therefore V~/(V~ + W,). These F centers in the state Ix)In - I, x) will again relax down to Ix)In - 2, x ) with probability Vn-1 or tunnel to Ig)ln - 1 + p, g) with probability Wn-l , and the process continues. Thus starting with the state Ix)lno, x), the fraction o f F centers which reach the state Ix)J0, x) and give rise to luminescence is given by
,7=~f n=l
v. v. + w,,
(5)
Energy
I
__t_ 8
A
....
L
Fig. 1. Configuration-coordinate diagram to illustrate the absorption and emission processes for F centers• We remark that we have neglected here the effect of the back transitions from the ground state to the excited state, e.g. Ig)ln + p, g) ~ Ix)In, x), also of probability Wn. We expect however that because n + p is in general very large (of the order of a hundred, see Table 1), the probability of intra-level transitions to lower vibrational levels is much larger than Wn when the system is on the ground state energy surface [3]. This point has also been discussed by Dexter and Fowler [6]. Thus equation (5) should be a good approximation. We now discuss the two probabilities Vn and W,,. According to Stoneham and Bartram [3] Vn = nTrA~
(6)
where Aco is a measure of the average coupling between Qeff and the lattice. Unfortunately, the value of Aco is difficult to obtain theoretically. Nevertheless a rough estimate of Aw may be made, since for most common defects Vn is of the order 101°-1012 sec -l [7]. For inter-level NRTs, we follow the expression of Pooley [8] which is an extension of the work of Huang and Rhys [9]. Confining to the case of zero temperature we obtain Wn = Wol(n, x l n + p,g)j2
(7)
where Wo is a physical factor involving the transition matrix element of the promoting mode between Ig) and Ix). The second factor is the square of the overlap integral of the two displaced harmonic oscillator wavefunctions (the Franck-Condon factor) which can readily be expressed in terms of the Laguerre polunomial L~ (S) [4, 10]. We have calculated Wn for various alkali halides.
Vol. 33, No. 8
ON THE LUMINESCENCE QUENCHING OF F CENTERS IN ALKALI HALIDES
For convenience the physical factor Wo is taken as 1013 sec-I [4, 11 ]. As expected our calculation shows that Wn is largest when n ~ n c , i.e. when the vibrational levels are close to the crossing point of the two potential curves. The largest value is about 2 x 1011 sec-i for all cases. As n decreases from n c , Wn decreases quite rapidly; at about 20 levels below n o Wn decreases by more than five orders of magnitude. For n >~ n c, I¢n oscillates within the range 1011-1012 sec-1 for at least some thirty to forty vibrational levels apart from some fluctuations (see Fig. 2 of [4] ). The important point here is that the probabilities of inter-level transitions from levels above the crossing point may be quite comparable to those from levels near the crossing point. We are now in a position to give a rough estimate of 7/. We consider first those alkali halides in Table 1 for which A ( ¼. For these alkali halides S ~ n c - 20 (see Table 1) which means that the level reached by optical absorption is at least some twenty levels below the crossing point. According to our calculation discussed above, the inter-level transition probability Wn is less than 10 7 sec -1 , which is much smaller than Vn(10 n ~ 1012 sec-1). Thus the quantum efficiency of luminescence 77 ~ 1. On the other hand for those alkali halides with A ) ¼, excluding NaC1 for the moment, the level reached by optical absorption is at least some thirty levels above the crossing point (S ~ n c + 30). Since Wn is quite comparable with V, for levels near or above the crossing point, we expect 7? to be quite small. As a rough estimate, we may assume Wn to be about 1011 sec -1 for twenty levels above or near n c and zero otherwise. Taking Vn as 1011-1012sec -1, we obtain from equation (5) 0 ~< 7? <~ 0.15. Thus for these alkali halides, luminescence is expected to have a low efficiency or even be completely quenched. Although our prediction is in general accora with that based on the DKR model, the present model in fact gives a more stringent criterion for the quenching of luminescence than that of the DKR model. For luminescence to be (almost) quenched, the present model requires not only S ) nc(i.e. A ) ¼) but that S be considerably larger than n c (e.g. S >~ n c + 30). Likewise for luminescence efficiency to be close to 1, S has to be considerably smaller than n c (e.g. S <~ n c + 20), as compared with S ( n c (i.e. A (¼) in the DKR model. In the case when S -~ n c , as in NaC1, the efficiency of luminescence can only be determined by detailed calculations based on equations (5)--(7). As a comparative study, we have calculated 7? for the alkali halides in Table 1 with A ) ¼ using equations (5)-(7). We have taken rrA6o to be 5 x 101o sec-1, and for convenience only up to forty
909
levels (n = 1 ~ 40) have been included in equation (5) in all cases. The results of this calculation are also presented in Table 1. The efficiency r/is only about a few % for these alkali halides except for LiI and NaCI, whose values of n c are relatively large. Luminescence of the F center in NaC1 has been observed with a quantum efficiency of 0.33 [12]. Bartram and Stoneham [3] suggested that luminescence in NaC1 is expected because 26% of the F band absorption can populate excited states with energy less than that of the crossing point. The present model provides an alternative explanation that because of the competition between inter-level and intra-level transitions as the system relaxes, luminescence efficiency may still be large ifS is larger than but close to n c . As for the case LiI, luminescence has not been observed. However, LiI is quite anomalous because the potential minimum of the excited state lies outside the potential well of the ground state (see Fig. 2 of [2] ). Thus even if r/is not small, luminescence may be quite difficult to observe. The present work may be considered as complementing the recent formulation of Bartram and Stoneham of the earlier idea of Dexter et al. In a very recent work, Kusunoki [13] has presented a quantitative theory of luminescence quenching which also takes explicit account of the coupling between Ix) and Ig) as well as the damping of the lattice vibration. Using the Landau-Zener theory of molecular collision, the author arrives at similar conclusion as we do here regarding the partial quenching of luminescence for cases with A >~ 0.25. A c k n o w l e d g e m e n t - This work was supported in part
by a grant from the National Science and Engineering Research Council of Canada. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
D.L. Dexter, C.C. Klick & G.A. Russel, Phys. Rev. 100,603 (1955). R.H. Bartram & A.M. Stoneham, Solid State C o m m u n . 17, 1593 (1975) A.M. Stoneham & R.H. Bartram, Solid S t a t e Electronics 21, 1325 (1978). K.S. Song& C.H. Leung, Solid State C o m m u n . 32, 565 (1979). R.K. Dawson& D. Pooley, Phys. Status Solidi 35, 95 (1969). D.L. Dexter & W.B. Fowler, J. Chem. Phys. 47, 1379 (1967). A.M. Stoneham, Theory o f Defects in Solids, p. 548. Oxford University Press (1975). D. Pooley, Proc. Phys. Soc. L o n d o n 87, 245 (1966). K. ttuang & A. Rhys, Proc. R . Soc. A 2 0 4 , 406 (1950).
910 10. 11.
ON THE LUMINESCENCE QUENCHING OF F CENTERS IN ALKALI HALIDES T.H. Keil, Phys. Rev. A140, 601 (1965). Y. Suzuki, M. Okumura & M. Hirai, J. Phys. Soc. Japan 47, 184 (1979).
12. 13.
Vol. 33, No. 8
S. Honda & M. Tomura, J. Phys, Soc. Japan 33, 1003 (1972). M. Kusunoki, Phys. Rev. B20, 2512 (1979).