- Email: [email protected]
Are there charged two-level systems in polymers?
21 August 1998
Chemical Physics Letters 293 Ž1998. 138–144
Are there charged two-level systems in polymers? I.S. Osad’ko ) , N.N. Zaitsev Department of Physics, Moscow State Pedagogical UniÕersity, 119435, Moscow, Russian Federation Received 28 April 1998; in final form 30 June 1998
Abstract It is shown that a square-root temporal hole broadening found recently in polymethylmethacrylate ŽPMMA. over a time scale ranging from 10 3 to 10 6 s can be explained in the framework of the standard two-level system model with an additional assumption that only a small part Ž; 10y11 . of the two-level system existing in PMMA has a charge. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The most intriguing phenomenon discovered in optical experiments with polymers and glasses doped with impurity centers is probably so-called spectral diffusion ŽSD.. SD manifests itself in spontaneous temporal broadening of spectral holes burnt in spectral bands w1,2x and in the dependence of the optical dephasing time T2 Ž t w . on the waiting time t w measured in stimulated photon echo experiments w3,4x. By now there are theories for SD of stochastic w5–9x and dynamic w10–13x type. Both theories explain SD as the result of an interaction between a chromophore and slowly relaxing so-called two-level systems ŽTLS., which were successfully invoked twenty-five years ago w14,15x to explain some low-temperature anomalies in glasses. All the theories for SD are based on two assumptions: Ž1. the distribution with respect to the TLS splitting E and rate constants R does not depend on E and has a hyperbolic character with respect to R: N Ž E, R . s
const. R
;
Ž 1.
and Ž2. the interaction between a chromophore and TLS is of a dipolar type. Although the stochastic and dynamic theories use different mathematical methods nevertheless both, as it was shown in Refs. w11,13x, predict a Lorentzian lineshape and yield similar expressions for the linewidth. If the hyperbolic distribution function N Ž E, R . is used, the linewidth must have a logarithmic temporal broadening. This temporal broadening was found in experiments w16x.
)
Corresponding author. E-mail: [email protected]
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 7 8 0 - 5
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
139
However, the predictions of the stochastic and dynamic theories differ considerably if the chromophore–TLS interaction differs from a dipolar one. The stochastic theory predicts a deviation from the Lorentzian lineshape in this case. In accordance with the dynamic theory, the lineshape does not depend on the type of the chromophore–TLS interaction and remains Lorentzian. Haarer and co-workers w17,18x have recently found that the prediction of the theories which concerns logarithmic temporal hole broadening is confirmed only over a time scale ranging from 1 to 10 3 s. As to broadening over a time scale ranging from 10 3 to 10 6 s, the hole broadening follows a power low t 1r2 . They could fit their experimental data with the help of stochastic SD theory with a modified distribution function 1
N Ž E, R . s const.
A q
R
R 3r2
.
Ž 2.
The additional term ArR 3r2 comes from the pair of TLS which interact with each other. This term is predicted in the first version of the theory by Burin and Kagan w19x. This theory allows for a dipolar interaction between TLS. The contribution to the distribution function N Ž E, R . from the pairs of interacting TLS is proportional to Ry3 r2 . However, recently new theoretical arguments and experimental facts which will be discussed in Section 3 appeared. They contradict the theoretical explanation given. Therefore, we undertake a reinvestigation of the experimental data by Haarer and co-workers beyond the frame of a stochastic approach to SD. We used the formulae of our dynamical theory for SD. We are able to explain the experimental data using the conventional distribution function described by Eq. Ž1. and an additional assumption that among the huge number of uncharged TLS with dipolar moments there are small number of charged TLS. Our assumption can be related to existing well-known facts of photoconductivity in pure polymers w20,21x.
2. Optical lineshape and linewidth The lineshape of a chromophore is determined as the Fourier transform of a dipolar correlator ²dŽ t .dŽ0.: s d 2 I Ž t .: `
IŽ v . s
Hy` I Ž t . e
ivt
dt .
Ž 3.
Neglecting the interaction between a chromophore and TLS, we find that I Ž t . A expŽyi v 0 t . and therefore I Ž v . A dŽ v y v 0 .. There is no line broadening in this case. Let us consider the influence of a chromophore–TLS interaction. The stochastic and dynamic theories do this in different fashion. 2.1. Stochastic approach Let a chromophore interact with a single TLS. Due to this interaction, the resonant frequency v 0 of the chromophore changes at time t w its value, which becomes v 0 q D. Then we find I Ž t . s eyi t v 0 1 y r Ž t w . q e i tŽ v 0q D .r Ž t w . ,
Ž 4.
where r Ž t w . is the probability of finding the new value of the frequency. If the chromophore interacts with N0 TLS we find, instead of Eq. Ž4., the following expression N0
I Ž t . s eyi t v 0 Ł 1 y r j Ž t w . Ž 1 y eyt D j . , j
Ž 5.
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
140
where D j is a line shift due to the interaction with the jth TLS. This equation is the starting point in stochastic SD theory. Let us introduce the distribution function 1 N Ž E, R . s Ž 6. Ý d Ž E y Ej . d Ž R y R j . . N0 j The frequency shift D j s DŽ r 0 j . is a function of the vector r 0 j connecting the impurity to the jth tunneling systems. Since Ej and R j are independent of r 0 j , the distribution functions with respect to D and the function N Ž E, R . can be treated as statistically independent. Using Eq. Ž5. we can write I Ž t w , t . s exp yi v 0 t q g Ž t w , t . ,
Ž 7.
where g Ž t w , t . s Ý ln 1 y r j Ž 1 y eyi D j t . s Hd R d E N Ž E, R . Ý ln 1 y r Ž E, R , t w . Ž 1 y eyi D j t . . j
Ž 8.
j
The total number N of lattice nodes is larger than the number N0 of TLS. Using a large number N we can replace the summation over tunneling systems with an integration over the coordinates. After that, the sum is transformed to the integral N0 Ý ln 1 y r Ž E, R . Ž 1 y eyi D j t . s V dV ln 1 y r Ž E, R , t w . Ž 1 y eyi D t . V j
H
fy
N0 V
r Ž E, R , t w .
yi D t
HVdV Ž 1 y e
.,
Ž 9.
where V is the sample volume. The behavior of the function g Ž t, t w . at large t determines the chromophore lineshape. This lineshape depends on the type of chromophore–TLS interaction. Assuming that this interaction is of a dipolar type so that Dd DŽ r . s 3 F Ž u , w . , Ž 10 . r where r is the distance between a chromophore and TLS, D is the dipole moment of the chromophore and d the dipole moment of the TLS. Substituting Eq. Ž10. into Eq. Ž9. and using a spherical coordinate system, we find for large t: N0 Dd Gs ² < F < :Hd E d R N Ž E, R . r Ž E, R , t w . s y< t < , Reg Ž t w , t . A y< t < Ž 11 . V 3 2 where ² . . . : denotes an integration with respect to angles. In this case the lineshape function I Ž v . is described by a Lorentzian. However, if D j A 1rr j2 or 1rr j6 , we obtain that Rew g Ž t, t w .x A t 3r2 or t 1r2 , respectively. In these cases the optical line is of non-Lorentzian type. Therefore the Lorentzian lineshape or purely exponential echo decay in photon echo experiments served in the past as an argument for the dipolar character of the chromophore–TLS interaction. Allowing for r Ž t w ., the following expression
r Ž t w . s f Ž E . Ž 1 y eyt w R . , Ž 12 . y1 Ž . w Ž . x Ž . where f E s exp ErkT q 1 , we obtain with the help of Eq. 11 the well-known formula for the linewidth w2–4x: Gs N0 Dd ² < F < :Hd E d R N Ž E, R . f Ž E . Ž 1 y eyi R t w . . s Ž 13 . 2 V 3 This equation was used by Haarer and co-workers to fit their experimental data w17,18x.
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
141
2.2. Dynamical approach In this approach the dynamics of the system is determined completely by its Hamiltonian. In order to calculate a dipolar correlator the quantum statistical average with a nonequilibrium density matrix for the TLS is used. In the dynamical theory, instead of Eq. Ž5., the following equation was derived w12,13x: N0
I Ž t , t w . s exp yi Ž v 0 q d . t y < t < Gdr2
Ł j
1 y rj
Dj Dj y i R j
1 y exp Žyi Dj y R j . t 4
,
Ž 14 .
where N0
ds Ý j
Gd 2
Dj R 2 D j2 q R 2j
N0
rj Ž1 y rj . ,
D j2 R j
sÝ
D j2 q R 2j
j
rj Ž1 y rj . .
Ž 15a. Ž 15b.
Here all r j s r j Ž t w .. Eq. Ž14. differs considerably from Eq. Ž5., which is the result of stochastic theory. However, all the differences disappear when R j s 0. But if R j s 0, we may not retain a temporal dependence in the functions r j Ž t w .. Therefore, Eq. Ž5. and hence Eq. Ž11. are not quite correct. Eq. Ž14. in contrast to Eq. Ž5. is correct at R j / 0. If R j / 0, the huge number of TLS which are situated far from the chromophore do not contribute to the product Ł j because D jrŽ D j y iR j . ™ 0 for these TLS. Therefore, in contrast to the stochastic theory where the product Ł plays the decisive role, the main contribution to the hole width in the dynamical theory comes from Gd . By neglecting r i2 in Eqs. Ž15a. and Ž15b. we can easily find:
Gd 2
N0
D j2 R j
fÝ
D j2 q R 2j
j
rj s
N0 V
Hd E d R N Ž E, R . Ž 1 y eyt w R . f Ž E . I Ž R . ,
Ž 16 .
where IŽ R. sR
HV D
D2 Ž r . 2
Ž r . q R2
dV .
Ž 17 .
Eq. Ž16. differs from Eq. Ž13. by the integral I Ž R .. It should note that Eq. Ž14. predicts a Lorentzian lineshape independently of the type of the chromophore–TLS interaction.
3. Discussion and comparison to experiment By substituting the function N Ž E, R . in Eq. Ž13. by Eq. Ž2. the authors of Refs. w17,18x used this expression to fit their experimental data depicted in Fig. 1. They found good agreement. It seemed that the interaction between the TLS can serve as the physical reason of the square-root temporal hole broadening found experimentally. However, if we take into account some recent experimental and theoretical findings, we find some sore points in the theoretical interpretation proposed by the Haarer group. The first objection is purely theoretical. The additional term in the distribution function N Ž E, R ., which is proportional to Ry3 r2 and which provides the square-root temporal behavior of the hole width exists in the first version of the theory by Burin and Kagan w19x. This theory predicts the appearance in a glass of the pairs of interacting TLS. However, in a more advanced theory developed recently w22x, Burin and Kagan have found that a dipolar interaction between TLS promotes an aggregation of the TLS. These aggregates can include two,
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
142
three, four, etc., TLS. Burin and Kagan calculated a contribution to the distribution function from aggregates of all the types. They have found that a new distribution function occurs, again of almost a hyperbolic type. In fact, they proved that the dipolar interaction between TLS does not change the hyperbolic distribution with respect to R. Therefore the second term in Eq. Ž2. is deprived of its theoretical justification. The second objection is based on the experimental data plotted in Fig. 1. The Haarer group chose the parameter A in Eq. Ž2. to fit their experimental data. The parameter A, which fits the data for tetra-phenyl-porphin ŽTPP. in polymethylmethacrylate ŽPMMA. is three times smaller compared to the A which fits the data for H 2 PC in PMMA. However, this parameter in the distribution function N Ž E, R . characterizes solely the solvent, i.e. PMMA. This parameter cannot depend on the type of guest molecule. It is possible to assume the parameter A to be temperature dependent and ascribe the decreasing of A to heating of the sample. However, then we need explain why parameter A becomes temperature independent for phthalocyanine ŽH 2 PC. in PMMA. Indeed one value of A fits data for 0.5 and 1 K well. Taking into account these objections we have to pass over Eq. Ž2. and try to find the another explanation for the intriguing experimental data by the Haarer group. We shall use our dynamic SD theory, which results in Eq. Ž16.. It was found that TLS have a dipole moment d f 0.4 debye w23x. The chromophore–TLS interaction is of dipole–dipole type so that D j s Ddrr j3 s D d, where D is a chromophore dipole moment. We assume that in parallel with the uncharged TLS, there is small amount of charged TLS with a chromophore–TLS interaction of monopole–dipole type: D j s Derr j2 s D e, where e is the charge of the TLS. In accordance with this assumption we have Žd.
Ý j
Ž e.
... s Ý ...qÝ ... , j
Ž 18.
j
where the index j runs over the uncharged and charged TLS, respectively. After we substitute the summation over the TLS by an integration over space, we come to Eq. Ž16. where I Ž R . s n d Id Ž R . q n e Ie Ž R . ,
Ž 19 .
where n d and n e are the concentrations of uncharged and charged TLS, respectively. The integrals Id Ž R . and
Fig. 1. Logarithmic and square-root temporal hole broadening. H 2 PC in PMMA at 0.05 K Žsquares. and 1 K Žcircles.; TPP in PMMA at 2 K Žtriangles.. The theoretical curves were calculated with the help of Eq. Ž23..
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
143
Ie Ž R . are described by Eq. Ž17., where D is substituted by D d s Ddrr 3 and D e s Derr 2 , respectively. By carrying out the integration we find p Id s Dd , Ž 20a. 6 Ie s Ž De .
1
3r2
R
1r2
.
Ž 20b.
We choose the distribution function in a conventional form: N Ž E, R . s N1Ž E . N2 Ž R .: N1 Ž E . s N2 Ž R . s
½
Ea , 0,
½
R ln Ž R 2rR1 . 0,
0 - Em , elsewhere , y1
5
,
Ž 21 . R1 - R 2 , elsewhere ,
5
Ž 22 .
Here 0 and Em or R 1 and R 2 are the boundaries of the distribution functions. By inserting Eq. Ž20a. into Eq. Ž16., we find the following expression for the linewidth
Gd s CT 1q a
R2
HR
dR
1
ž
1 q R
6 n g ´ 1r2 p n d R 3r2
/
Ž 1 y eyt w R . f CT 1q a ln t q
12 n g p nd
´
ž / R2
1r2
't
,
Ž 23 .
where Cs
kB
ž / Em
1q a
p
´ l3
3 ln Ž R 2rR1 .
nd .
Ž 24 .
Here t s R 2 t w , ´ s Ddrl 3 s Derl 2 and d s el. Using Eq. Ž23. we calculated the theoretical curves which are plotted in Fig. 1. They fit the experimental data well. The shape of the theoretical curves depends on three parameters: ´ , R 2 and the ratio n ern d of charged to uncharged TLS. We assume that D s d s 0.4 debye and e is the electronic charge. Then ´ f 12 Ry f 150 eV f 4 = 10 16 sy1 . The theoretical curves fit the experimental data at C s 4.1 MHzrK, a s 0.55, R 2 s 10 11 s and n ern d s 2.5 = 10y1 1. The ratio of charged to uncharged TLS is small. It seems that so small a number of charged TLS cannot be detected by means of non-optical methods. Experimental data for H 2 PC in PMMA were described by one set of parameters C, a , R 2 , n ern d and ´ . For TPP in PMMA we choose the parameter ´ to be 1.25 times smaller. This parameter includes the dipole moment of the chromophore. All parameters are temperature independent. It is worth noting that the square-root temporal dependence in our theory results from the integral Ie Ž R . in the equation for linewidth. Therefore our theory cannot explain the appearance of the square-root temporal dependence in non-optical experiments. However, the distribution function described by Eq. Ž2. must manifest itself in the low-temperature experiments of various kinds.
4. Conclusions We have shown that the square-root temporal behavior of the hole width, which had been recently discovered in PMMA over an ultralong time scale can be explained without invoking an interaction between the TLS and a revision of the so-called standard TLS model. Our explanation is based on dynamical SD theory and the assumption concerning the existence of a small amount of charged TLS in PMMA.
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
144
Acknowledgements This work was supported by Russian Foundation for Basic Research grant 97-02-17285.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x
W. Breinl, J. Friedrich, D. Haarer, J. Chem. Phys. 81 Ž1984. 3915. K. Littau, Y.S. Bay, M.D. Fayer, J. Chem. Phys. 92 Ž1990. 4145. H.C. Meijers, D.A. Wiersma, Phys. Rev. Lett. 68 Ž1992. 381. H.C. Meijers, D.A. Wiersma, J. Chem. Phys. 101 Ž1994. 6927. R. Klauder, P.W. Anderson, Phys. Rev. 125 Ž1962. 912. P. Hu, S.B. Hartmann, Phys. Rev. B 9 Ž1974. 1. P. Hu, L.R. Walker, Phys. Rev. B 18 Ž1978. 1300. Y.S. Bay, M.D. Fayer, Phys. Rev. B 36 Ž1989. 11066. P.D. Reilly, J.L. Skinner, J. Chem. Phys., 101 Ž1994. 959, 965. I.S. Osad’ko, JETP Lett. 61 Ž1995. 270. I.S. Osad’ko, M.A. Mikhailov, Phys. Lett. A 208 Ž1995. 231. I.S. Osad’ko, N.N. Zaitsev, Chem. Phys. Lett. 257 Ž1996. 82. I.S. Osad’ko, JETP 82 Ž1996. 434. P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag. 25 Ž1972. 1. W.A. Phillips, J. Low Temp. Phys. 7 Ž1972. 351. W. Kohler, J. Meiler, J. Friedrich, Phys. Rev. B 35 Ž1987. 4031. H. Maier, B.M. Kharlamov, D. Haarer, Phys. Rev. Lett. 76 Ž1996. 2085. G. Hannig, H. Maier, D. Haarer, B.M. Kharlamov, Mol. Cryst.-Liq. Cryst. 291 Ž1996. 11. A.L. Burin, Yu. Kagan, JETP 79 Ž1994. 347. H. Scher, E.W. Montroll, Phys. Rev. B 12 Ž1975. 2455. E. Muller-Horsche, D. Haarer, H. Scher, Phys. Rev. B 35 Ž1987. 1273. A.L. Burin, Yu. Kagan, JETP 82 Ž1996. 159. H. Maier, R. Wunderlich, D. Haarer, B.M. Kharlamov, S.G. Kulikov, Phys. Rev. Lett. 74 Ž1995. 5252.
Chemical Physics Letters 293 Ž1998. 138–144
Are there charged two-level systems in polymers? I.S. Osad’ko ) , N.N. Zaitsev Department of Physics, Moscow State Pedagogical UniÕersity, 119435, Moscow, Russian Federation Received 28 April 1998; in final form 30 June 1998
Abstract It is shown that a square-root temporal hole broadening found recently in polymethylmethacrylate ŽPMMA. over a time scale ranging from 10 3 to 10 6 s can be explained in the framework of the standard two-level system model with an additional assumption that only a small part Ž; 10y11 . of the two-level system existing in PMMA has a charge. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The most intriguing phenomenon discovered in optical experiments with polymers and glasses doped with impurity centers is probably so-called spectral diffusion ŽSD.. SD manifests itself in spontaneous temporal broadening of spectral holes burnt in spectral bands w1,2x and in the dependence of the optical dephasing time T2 Ž t w . on the waiting time t w measured in stimulated photon echo experiments w3,4x. By now there are theories for SD of stochastic w5–9x and dynamic w10–13x type. Both theories explain SD as the result of an interaction between a chromophore and slowly relaxing so-called two-level systems ŽTLS., which were successfully invoked twenty-five years ago w14,15x to explain some low-temperature anomalies in glasses. All the theories for SD are based on two assumptions: Ž1. the distribution with respect to the TLS splitting E and rate constants R does not depend on E and has a hyperbolic character with respect to R: N Ž E, R . s
const. R
;
Ž 1.
and Ž2. the interaction between a chromophore and TLS is of a dipolar type. Although the stochastic and dynamic theories use different mathematical methods nevertheless both, as it was shown in Refs. w11,13x, predict a Lorentzian lineshape and yield similar expressions for the linewidth. If the hyperbolic distribution function N Ž E, R . is used, the linewidth must have a logarithmic temporal broadening. This temporal broadening was found in experiments w16x.
)
Corresponding author. E-mail: [email protected]
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 7 8 0 - 5
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
139
However, the predictions of the stochastic and dynamic theories differ considerably if the chromophore–TLS interaction differs from a dipolar one. The stochastic theory predicts a deviation from the Lorentzian lineshape in this case. In accordance with the dynamic theory, the lineshape does not depend on the type of the chromophore–TLS interaction and remains Lorentzian. Haarer and co-workers w17,18x have recently found that the prediction of the theories which concerns logarithmic temporal hole broadening is confirmed only over a time scale ranging from 1 to 10 3 s. As to broadening over a time scale ranging from 10 3 to 10 6 s, the hole broadening follows a power low t 1r2 . They could fit their experimental data with the help of stochastic SD theory with a modified distribution function 1
N Ž E, R . s const.
A q
R
R 3r2
.
Ž 2.
The additional term ArR 3r2 comes from the pair of TLS which interact with each other. This term is predicted in the first version of the theory by Burin and Kagan w19x. This theory allows for a dipolar interaction between TLS. The contribution to the distribution function N Ž E, R . from the pairs of interacting TLS is proportional to Ry3 r2 . However, recently new theoretical arguments and experimental facts which will be discussed in Section 3 appeared. They contradict the theoretical explanation given. Therefore, we undertake a reinvestigation of the experimental data by Haarer and co-workers beyond the frame of a stochastic approach to SD. We used the formulae of our dynamical theory for SD. We are able to explain the experimental data using the conventional distribution function described by Eq. Ž1. and an additional assumption that among the huge number of uncharged TLS with dipolar moments there are small number of charged TLS. Our assumption can be related to existing well-known facts of photoconductivity in pure polymers w20,21x.
2. Optical lineshape and linewidth The lineshape of a chromophore is determined as the Fourier transform of a dipolar correlator ²dŽ t .dŽ0.: s d 2 I Ž t .: `
IŽ v . s
Hy` I Ž t . e
ivt
dt .
Ž 3.
Neglecting the interaction between a chromophore and TLS, we find that I Ž t . A expŽyi v 0 t . and therefore I Ž v . A dŽ v y v 0 .. There is no line broadening in this case. Let us consider the influence of a chromophore–TLS interaction. The stochastic and dynamic theories do this in different fashion. 2.1. Stochastic approach Let a chromophore interact with a single TLS. Due to this interaction, the resonant frequency v 0 of the chromophore changes at time t w its value, which becomes v 0 q D. Then we find I Ž t . s eyi t v 0 1 y r Ž t w . q e i tŽ v 0q D .r Ž t w . ,
Ž 4.
where r Ž t w . is the probability of finding the new value of the frequency. If the chromophore interacts with N0 TLS we find, instead of Eq. Ž4., the following expression N0
I Ž t . s eyi t v 0 Ł 1 y r j Ž t w . Ž 1 y eyt D j . , j
Ž 5.
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
140
where D j is a line shift due to the interaction with the jth TLS. This equation is the starting point in stochastic SD theory. Let us introduce the distribution function 1 N Ž E, R . s Ž 6. Ý d Ž E y Ej . d Ž R y R j . . N0 j The frequency shift D j s DŽ r 0 j . is a function of the vector r 0 j connecting the impurity to the jth tunneling systems. Since Ej and R j are independent of r 0 j , the distribution functions with respect to D and the function N Ž E, R . can be treated as statistically independent. Using Eq. Ž5. we can write I Ž t w , t . s exp yi v 0 t q g Ž t w , t . ,
Ž 7.
where g Ž t w , t . s Ý ln 1 y r j Ž 1 y eyi D j t . s Hd R d E N Ž E, R . Ý ln 1 y r Ž E, R , t w . Ž 1 y eyi D j t . . j
Ž 8.
j
The total number N of lattice nodes is larger than the number N0 of TLS. Using a large number N we can replace the summation over tunneling systems with an integration over the coordinates. After that, the sum is transformed to the integral N0 Ý ln 1 y r Ž E, R . Ž 1 y eyi D j t . s V dV ln 1 y r Ž E, R , t w . Ž 1 y eyi D t . V j
H
fy
N0 V
r Ž E, R , t w .
yi D t
HVdV Ž 1 y e
.,
Ž 9.
where V is the sample volume. The behavior of the function g Ž t, t w . at large t determines the chromophore lineshape. This lineshape depends on the type of chromophore–TLS interaction. Assuming that this interaction is of a dipolar type so that Dd DŽ r . s 3 F Ž u , w . , Ž 10 . r where r is the distance between a chromophore and TLS, D is the dipole moment of the chromophore and d the dipole moment of the TLS. Substituting Eq. Ž10. into Eq. Ž9. and using a spherical coordinate system, we find for large t: N0 Dd Gs ² < F < :Hd E d R N Ž E, R . r Ž E, R , t w . s y< t < , Reg Ž t w , t . A y< t < Ž 11 . V 3 2 where ² . . . : denotes an integration with respect to angles. In this case the lineshape function I Ž v . is described by a Lorentzian. However, if D j A 1rr j2 or 1rr j6 , we obtain that Rew g Ž t, t w .x A t 3r2 or t 1r2 , respectively. In these cases the optical line is of non-Lorentzian type. Therefore the Lorentzian lineshape or purely exponential echo decay in photon echo experiments served in the past as an argument for the dipolar character of the chromophore–TLS interaction. Allowing for r Ž t w ., the following expression
r Ž t w . s f Ž E . Ž 1 y eyt w R . , Ž 12 . y1 Ž . w Ž . x Ž . where f E s exp ErkT q 1 , we obtain with the help of Eq. 11 the well-known formula for the linewidth w2–4x: Gs N0 Dd ² < F < :Hd E d R N Ž E, R . f Ž E . Ž 1 y eyi R t w . . s Ž 13 . 2 V 3 This equation was used by Haarer and co-workers to fit their experimental data w17,18x.
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
141
2.2. Dynamical approach In this approach the dynamics of the system is determined completely by its Hamiltonian. In order to calculate a dipolar correlator the quantum statistical average with a nonequilibrium density matrix for the TLS is used. In the dynamical theory, instead of Eq. Ž5., the following equation was derived w12,13x: N0
I Ž t , t w . s exp yi Ž v 0 q d . t y < t < Gdr2
Ł j
1 y rj
Dj Dj y i R j
1 y exp Žyi Dj y R j . t 4
,
Ž 14 .
where N0
ds Ý j
Gd 2
Dj R 2 D j2 q R 2j
N0
rj Ž1 y rj . ,
D j2 R j
sÝ
D j2 q R 2j
j
rj Ž1 y rj . .
Ž 15a. Ž 15b.
Here all r j s r j Ž t w .. Eq. Ž14. differs considerably from Eq. Ž5., which is the result of stochastic theory. However, all the differences disappear when R j s 0. But if R j s 0, we may not retain a temporal dependence in the functions r j Ž t w .. Therefore, Eq. Ž5. and hence Eq. Ž11. are not quite correct. Eq. Ž14. in contrast to Eq. Ž5. is correct at R j / 0. If R j / 0, the huge number of TLS which are situated far from the chromophore do not contribute to the product Ł j because D jrŽ D j y iR j . ™ 0 for these TLS. Therefore, in contrast to the stochastic theory where the product Ł plays the decisive role, the main contribution to the hole width in the dynamical theory comes from Gd . By neglecting r i2 in Eqs. Ž15a. and Ž15b. we can easily find:
Gd 2
N0
D j2 R j
fÝ
D j2 q R 2j
j
rj s
N0 V
Hd E d R N Ž E, R . Ž 1 y eyt w R . f Ž E . I Ž R . ,
Ž 16 .
where IŽ R. sR
HV D
D2 Ž r . 2
Ž r . q R2
dV .
Ž 17 .
Eq. Ž16. differs from Eq. Ž13. by the integral I Ž R .. It should note that Eq. Ž14. predicts a Lorentzian lineshape independently of the type of the chromophore–TLS interaction.
3. Discussion and comparison to experiment By substituting the function N Ž E, R . in Eq. Ž13. by Eq. Ž2. the authors of Refs. w17,18x used this expression to fit their experimental data depicted in Fig. 1. They found good agreement. It seemed that the interaction between the TLS can serve as the physical reason of the square-root temporal hole broadening found experimentally. However, if we take into account some recent experimental and theoretical findings, we find some sore points in the theoretical interpretation proposed by the Haarer group. The first objection is purely theoretical. The additional term in the distribution function N Ž E, R ., which is proportional to Ry3 r2 and which provides the square-root temporal behavior of the hole width exists in the first version of the theory by Burin and Kagan w19x. This theory predicts the appearance in a glass of the pairs of interacting TLS. However, in a more advanced theory developed recently w22x, Burin and Kagan have found that a dipolar interaction between TLS promotes an aggregation of the TLS. These aggregates can include two,
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
142
three, four, etc., TLS. Burin and Kagan calculated a contribution to the distribution function from aggregates of all the types. They have found that a new distribution function occurs, again of almost a hyperbolic type. In fact, they proved that the dipolar interaction between TLS does not change the hyperbolic distribution with respect to R. Therefore the second term in Eq. Ž2. is deprived of its theoretical justification. The second objection is based on the experimental data plotted in Fig. 1. The Haarer group chose the parameter A in Eq. Ž2. to fit their experimental data. The parameter A, which fits the data for tetra-phenyl-porphin ŽTPP. in polymethylmethacrylate ŽPMMA. is three times smaller compared to the A which fits the data for H 2 PC in PMMA. However, this parameter in the distribution function N Ž E, R . characterizes solely the solvent, i.e. PMMA. This parameter cannot depend on the type of guest molecule. It is possible to assume the parameter A to be temperature dependent and ascribe the decreasing of A to heating of the sample. However, then we need explain why parameter A becomes temperature independent for phthalocyanine ŽH 2 PC. in PMMA. Indeed one value of A fits data for 0.5 and 1 K well. Taking into account these objections we have to pass over Eq. Ž2. and try to find the another explanation for the intriguing experimental data by the Haarer group. We shall use our dynamic SD theory, which results in Eq. Ž16.. It was found that TLS have a dipole moment d f 0.4 debye w23x. The chromophore–TLS interaction is of dipole–dipole type so that D j s Ddrr j3 s D d, where D is a chromophore dipole moment. We assume that in parallel with the uncharged TLS, there is small amount of charged TLS with a chromophore–TLS interaction of monopole–dipole type: D j s Derr j2 s D e, where e is the charge of the TLS. In accordance with this assumption we have Žd.
Ý j
Ž e.
... s Ý ...qÝ ... , j
Ž 18.
j
where the index j runs over the uncharged and charged TLS, respectively. After we substitute the summation over the TLS by an integration over space, we come to Eq. Ž16. where I Ž R . s n d Id Ž R . q n e Ie Ž R . ,
Ž 19 .
where n d and n e are the concentrations of uncharged and charged TLS, respectively. The integrals Id Ž R . and
Fig. 1. Logarithmic and square-root temporal hole broadening. H 2 PC in PMMA at 0.05 K Žsquares. and 1 K Žcircles.; TPP in PMMA at 2 K Žtriangles.. The theoretical curves were calculated with the help of Eq. Ž23..
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
143
Ie Ž R . are described by Eq. Ž17., where D is substituted by D d s Ddrr 3 and D e s Derr 2 , respectively. By carrying out the integration we find p Id s Dd , Ž 20a. 6 Ie s Ž De .
1
3r2
R
1r2
.
Ž 20b.
We choose the distribution function in a conventional form: N Ž E, R . s N1Ž E . N2 Ž R .: N1 Ž E . s N2 Ž R . s
½
Ea , 0,
½
R ln Ž R 2rR1 . 0,
0 - Em , elsewhere , y1
5
,
Ž 21 . R1 - R 2 , elsewhere ,
5
Ž 22 .
Here 0 and Em or R 1 and R 2 are the boundaries of the distribution functions. By inserting Eq. Ž20a. into Eq. Ž16., we find the following expression for the linewidth
Gd s CT 1q a
R2
HR
dR
1
ž
1 q R
6 n g ´ 1r2 p n d R 3r2
/
Ž 1 y eyt w R . f CT 1q a ln t q
12 n g p nd
´
ž / R2
1r2
't
,
Ž 23 .
where Cs
kB
ž / Em
1q a
p
´ l3
3 ln Ž R 2rR1 .
nd .
Ž 24 .
Here t s R 2 t w , ´ s Ddrl 3 s Derl 2 and d s el. Using Eq. Ž23. we calculated the theoretical curves which are plotted in Fig. 1. They fit the experimental data well. The shape of the theoretical curves depends on three parameters: ´ , R 2 and the ratio n ern d of charged to uncharged TLS. We assume that D s d s 0.4 debye and e is the electronic charge. Then ´ f 12 Ry f 150 eV f 4 = 10 16 sy1 . The theoretical curves fit the experimental data at C s 4.1 MHzrK, a s 0.55, R 2 s 10 11 s and n ern d s 2.5 = 10y1 1. The ratio of charged to uncharged TLS is small. It seems that so small a number of charged TLS cannot be detected by means of non-optical methods. Experimental data for H 2 PC in PMMA were described by one set of parameters C, a , R 2 , n ern d and ´ . For TPP in PMMA we choose the parameter ´ to be 1.25 times smaller. This parameter includes the dipole moment of the chromophore. All parameters are temperature independent. It is worth noting that the square-root temporal dependence in our theory results from the integral Ie Ž R . in the equation for linewidth. Therefore our theory cannot explain the appearance of the square-root temporal dependence in non-optical experiments. However, the distribution function described by Eq. Ž2. must manifest itself in the low-temperature experiments of various kinds.
4. Conclusions We have shown that the square-root temporal behavior of the hole width, which had been recently discovered in PMMA over an ultralong time scale can be explained without invoking an interaction between the TLS and a revision of the so-called standard TLS model. Our explanation is based on dynamical SD theory and the assumption concerning the existence of a small amount of charged TLS in PMMA.
I.S. Osad’ko, N.N. ZaitseÕr Chemical Physics Letters 293 (1998) 138–144
144
Acknowledgements This work was supported by Russian Foundation for Basic Research grant 97-02-17285.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x
W. Breinl, J. Friedrich, D. Haarer, J. Chem. Phys. 81 Ž1984. 3915. K. Littau, Y.S. Bay, M.D. Fayer, J. Chem. Phys. 92 Ž1990. 4145. H.C. Meijers, D.A. Wiersma, Phys. Rev. Lett. 68 Ž1992. 381. H.C. Meijers, D.A. Wiersma, J. Chem. Phys. 101 Ž1994. 6927. R. Klauder, P.W. Anderson, Phys. Rev. 125 Ž1962. 912. P. Hu, S.B. Hartmann, Phys. Rev. B 9 Ž1974. 1. P. Hu, L.R. Walker, Phys. Rev. B 18 Ž1978. 1300. Y.S. Bay, M.D. Fayer, Phys. Rev. B 36 Ž1989. 11066. P.D. Reilly, J.L. Skinner, J. Chem. Phys., 101 Ž1994. 959, 965. I.S. Osad’ko, JETP Lett. 61 Ž1995. 270. I.S. Osad’ko, M.A. Mikhailov, Phys. Lett. A 208 Ž1995. 231. I.S. Osad’ko, N.N. Zaitsev, Chem. Phys. Lett. 257 Ž1996. 82. I.S. Osad’ko, JETP 82 Ž1996. 434. P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag. 25 Ž1972. 1. W.A. Phillips, J. Low Temp. Phys. 7 Ž1972. 351. W. Kohler, J. Meiler, J. Friedrich, Phys. Rev. B 35 Ž1987. 4031. H. Maier, B.M. Kharlamov, D. Haarer, Phys. Rev. Lett. 76 Ž1996. 2085. G. Hannig, H. Maier, D. Haarer, B.M. Kharlamov, Mol. Cryst.-Liq. Cryst. 291 Ž1996. 11. A.L. Burin, Yu. Kagan, JETP 79 Ž1994. 347. H. Scher, E.W. Montroll, Phys. Rev. B 12 Ž1975. 2455. E. Muller-Horsche, D. Haarer, H. Scher, Phys. Rev. B 35 Ž1987. 1273. A.L. Burin, Yu. Kagan, JETP 82 Ž1996. 159. H. Maier, R. Wunderlich, D. Haarer, B.M. Kharlamov, S.G. Kulikov, Phys. Rev. Lett. 74 Ž1995. 5252.