solidi status physia pss urrent topis in solid state physis The role of breathers in the anomalous deay of luminesene Eva Miho kova and Lawrene S. Shulman Institute of Physis, ASCR, Cukrovarnika, Prague 6, 6 53, Czeh Republi Physis Department, Clarkson University, Potsdam, NY, 3699-58, USA Reeived 7 June 6, revised July 6, aepted July 6 Published online November 6 PACS 5.45.Yv, 3.7.Hq, 63..Ry, 78.55. m Luminesene of alkali halides doped by heavy ns ions exhibits an anomaly in the slow omponent emission deay. The anomaly is explained by the formation of a disrete breather in the immediate neighborhood of the impurity. We study properties of these breathers, their phase spae struture, robustness, and propensity for formation. Under a wide range of parameters and interioni potentials they form -dimensional Kolmogorov-Arnold-Moser tori (less than generi) in phase spae. We show strobed views of these tori, useful in quantization. All features support the thesis of breather formation as the explanation for the luminesene deay anomaly that first motivated our breather proposal. phys. stat. sol. () 3, No., 346 349 (6) / DOI./pss.674 T N I R P E R
phys. stat. sol. () 3, No., 346 349 (6) / DOI./pss.674 The role of breathers in the anomalous deay of luminesene Eva Mihóková and Lawrene S. Shulman Institute of Physis, ASCR, Cukrovarnika, Prague 6, 6 53, Czeh Republi Physis Department, Clarkson University, Potsdam, NY, 3699-58, USA Reeived 7 June 6, revised July 6, aepted July 6 Published online November 6 PACS 5.45.Yv, 3.7.Hq, 63..Ry, 78.55. m Luminesene of alkali halides doped by heavy ns ions exhibits an anomaly in the slow omponent emission deay. The anomaly is explained by the formation of a disrete breather in the immediate neighborhood of the impurity. We study properties of these breathers, their phase spae struture, robustness, and propensity for formation. Under a wide range of parameters and interioni potentials they form -dimensional Kolmogorov-Arnold-Moser tori (less than generi) in phase spae. We show strobed views of these tori, useful in quantization. All features support the thesis of breather formation as the explanation for the luminesene deay anomaly that first motivated our breather proposal. 6 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Introdution Breathers are spatially loalized, time-periodi, stable exitations in periodi nonlinear latties. They are known to exist in a number of systems, suh as Josephson juntions, optial waveguide arrays and laser-indued photoni rystals (see, e.g., [] and Refs. therein). In a series of studies in doped alkali halides [ 5] the following anomaly was found: rapid and non-exponential deay of luminesene during what should have been the early stages of the deay of a metastable level. This anomaly was explained by postulating extremely long rystal relaxation times. The extreme slowdown in turn was attributed to the formation of breathers in a lattie that is highly distorted by the Jahn-Teller (JT) effet [6]. The onnetion was established by simulating the dynamis of a hain of atoms along the JT axis using a quarti polynomial interatomi potential. In this paper we examine the effets of using different interatomi potentials as well as different atomi speies (in partiular different mass ratios for diatomi hains). We onfine attention to the zero-temperature ase. An interesting result is that, at least for the parameter values we study, the phase spae lassial motion of the atoms lives on a two-dimensional Kolmogorov-Arnold- Moser (KAM) torus. That is, beyond the already partiular property of regular, torus-onfined motion, that torus only has two signifiantly nonzero radii, i.e., nonzero ation variables. These results support, both physially and mathematially, the onlusions of [6]. The model and the breather in various interatomi potentials Optial properties of isolated Tl or Pb enters in alkali halide rystals an be desribed by onsidering the impurity and its six nearest neighbors (in the f lattie) to be a quasimoleule. When exited, the moleule distorts aording to the Jahn-Teller (JT) effet and its lowest exited level (whih was degenerate) is split into a radiative state, displaying exponential nanoseond deay and a metastable state, with a milliseond time sale. The latter is the state for whih the anomaly is observed. The initial deay is faster than exponential, settling into exponential deay only after a time interval on the order of milliseonds. To explain the anomaly we proposed that the lattie takes a long time to yield to the strain, thereby introduing a oupling between the two lowest exited states of the emission enter. This assumption allowed suessful fitting of low Corresponding author: e-mail: mihokova@fzu.z, Phone: +4 38 56, Fax: +4 33 343 84 6 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
phys. stat. sol. () 3, No. (6) 347 b.8.4 log Intensity 3 3 4.4 4 4 6 8 Frequeny.8..4.6 Fig. Quarti polynomial interatomi potential. a) Kineti energy of the system as a funtion of time and atom number. The inset shows the positions of the atoms vs. time. b) Fourier transforms (intensity) of the funtions (Q (t),q (t),q (t),q (t)) in urves through 4. The urves orresponding to suessive atoms are vertially shifted (downward by 4) for better display. The superimposed horizontal lines indiate where the phonon frequeny bands are. Arrows point to the prinipal frequenies of the spetrum outside the phonon bands. All other intense frequenies are resonanes or beat frequenies of these two. () The KAM torus, projeted on a plane within the four dimensional spae {Q,P,q,p }. Eah figure ontains points, with points taken per system time unit. The absissa is position and the ordinate is momentum. Loops in the torus represent projetions on the Q -P and q -p planes of the strobosopially viewed KAM torus, substantiating the assertion that the torus is two-dimensional (only has two signifiant nonzero radii). The parameters used in the simulation are N =, q =, r =, ν =4, and λ =. temperature KBr:Pb data [3], provided that the ultimate relaxation of the lattie is on the same time sale as that of the slow emission deay (ms). The model was extended to the entire olletion of Pb and Tldoped alkali halides both at liquid He temperature [4] and at higher temperatures [5]. Through modeling the dynamis of the lattie [6] we found that under lassial dynamis the energy of the extended quasi-moleule was onfined to a small neighborhood of the impurity-exitation-induedperturbation, a perturbation signifiant enough to make non-linear effets important. This onfinement led to the slowdown in relaxation needed to aount for the experimental data. We assume that the lattie anion immediately next to the impurity and along the axis of deformation is under tremendous pressure to move away from the impurity. We further assume that in studying the effet of this fore we an onfine attention to the line of atoms on this axis, effetively a hain, with the influene of the rest of the rystal expressed through a holding fore. We introdue this substrate-defining fore through a potential felt by eah ion on the hain; it defines the equilibrium position of the ion in the rystal in the absene of the speifi pressure from the displaed first anion. Note that the non-isotropi stresses imply lesser deformation for off-axis atoms. We assume an interioni potential V (u) with u the displaement ( from the ion s equilibrium position. For the simulations reported in [6] we used V (u) =Mω u + λu 4) /. The Hamiltonian is H = N n= { } P n M + rp n M + V (q n Q n )+V(Q n q n )+ν[v(q n )+V(q n )]. () Q, P are the oordinate and momentum of the anion to the right of the impurity, followed by q, p (ation), Q, P et. The Q-partiles have mass M, the q s, M/r ( r is a mass ratio). The holding fore arises from the potentials multiplying ν, so that ν an be interpreted as an effetive number of neighbors. This Hamiltonian inludes the effet of the highly distorted impurity wave funtion through the non-dynami variable q. By setting this to speifi positive values we an provide a push on the entire hain, induing intense osillation, possible breather formation, and possible wave propagation, depending on the model parameters. The system was solved numerially for the indiated potential, using units suh that M =and ω =. We found that for ertain parameters the energy deposited in the hain by the non-zero q (JT deformation) 6 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
348 E. Mihóková and L. S. Shuman: The role of breathers in the anomalous deay of luminesene b logintensity.4 3 4 3 5 Frequeny 5.4....3.4 Fig. Born-Mayer-Coulomb (BMC) potential: a), b), and ) as in Fig.. Parameters used in the simulation: N =, q =, r =, ν = 4. BMC parameters: αm q /d =, ρ =.3, and d = 3.3, in the notation of [7]. does not propagate but beomes bound in vibrations of the first atoms of the hain, i.e., the breather. This is illustrated in Fig. a displaying kineti energy as a funtion of time and atom number as well as the time dependene of the atomi position oordinate (in the inset). The frequeny spetrum of the first four atoms, obtained by Fast Fourier Transform (FFT) of Q (t), q (t), Q (t) q (t), is displayed in Fig. b. The spetrum shows two dominant frequenies falling above the optial phonon band and in its gap (marked by a sequene of horizontal lines). These frequenies haraterize the osillations of the breather. In Fig. we show a projetion of the KAM torus whih is the lassial phase spae orbit of the breather. Superimposed is a pair of loops, whih result from a strobosopi image of the torus [8]. The fat that these are well determined, one-dimensional loops implies that the KAM torus for our simulations is only a two dimensional struture. This is of ourse onsistent with the presene of only two dominant frequenies outside the phonon bands in the frequeny spetrum although there is no requirement that only two frequenies appear. There are many potentials used for the desription of interatomi fores. Those most frequently used are listed in [7]. Our onern was to study whether the breather is robust with respet to the hoie of interatomi potential. We alulated the hain dynamis for polynomial potential with a ubi term added, for Morse and for Born-Mayer-Coulomb potentials. All of them are softer than the originally used quarti polynomial potential. However, in all ases studied the formation of the breather was onfirmed. For illustration, the results for the Born-Mayer-Coulomb potential are shown in Fig.. 3 Preditions of the model in variety of substanes The deay anomaly has been observed in a variety of doped alkali halides. For potassium halides doped either with Tl+ or Pb+ the anomaly grows with inreasing size and mass of the lattie anion (a large anomaly is manifest for about 5 ms and enhanes deay by or 3 orders of magnitude). This ours in the sequene of latties KCl KBr KI. For the KCl lattie the anomaly, i.e. deviation from the exponential, is small (Tl+ ) or nonexistent (Pb+ ); for KBr the nonexponential part is steeper and survives to longer times for the KI survives even longer. Experimental results with Pb+ -doped alkali bromide rystals show that similar behavior is observed in the sequene of latties RbBr KBr NaBr where the lattie ation is suessively made smaller and lighter. Namely, for the RbBr rystal a weak anomaly is observed while there is a signifiant anomaly for Pb-doped KBr and NaBr rystals. To summarize, a quantity that ertainly affets the harater of the deay anomaly is, r, the ratio of the mass of the lattie anion to that of the lattie ation. In the sequene of potassium halides (KCl KBr KI) the orresponding mass ratio hanges as r to r to r 3.3. In the sequene of alkali bromides (RbBr KBr NaBr) the mass ratio goes from r to r to r 4. From the experimental point of view a very slight anomaly is observed or none at all for mass ratios lose to, while for ratios of or more the anomaly is large. How is this observation onsistent with our breather model? 6 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
phys. stat. sol. () 3, No. (6) 349 Kineti energy weighted position 5 5 5.5.5 3 r (mass ratio) Fig. 3 Kineti energy-weighted position (in atomi spaing units) as a funtion of anion-ation mass ratio. For details see the text. The parameters used in the simulation are N =, q =.5, ν =4, and λ =.5. To study this we used our standard quarti polynomial potential. We performed lattie-dynamis simulations hanging the atomi mass ratio entering the hain model. It turns that different mass ratios an be more favorable or less favorable for forming the breather. The simulations were made for a hain of 4 atoms. As a measure of how good the breather is, whih is to say, how good is the onfinement of the energy to the region of the impurity, we alulated a kineti energy-weighted position. That is, let k be the atom number, k =for the immediate neighbor of the impurity, and ounting outward. Let w KE (k) be the average kineti energy of the k th atom. Then k KE k = k kw KE(k) k w KE(k). () The smaller this quantity is, the more the energy is onentrated in the breather, and therefore the bigger the expeted deay anomaly. The result of a systemati study of k KE as a funtion of r is shown in Fig. 3. A mass ratio higher than is favorable for forming a breather and most of the kineti energy is trapped in first atoms of the hain. For mass ratios near one most of kineti energy travels away. This is ompletely and satisfyingly onsistent with the experimental observations. 4 Conlusion For doped alkali halides, the existene of breather modes provides an answer to the puzzling anomaly observed in slow omponent emission deay. First we found that breathers formed when the interatomi potential was quarti. In the present paper breather formation has been demonstrated for other forms of interatomi potential. Thus we onlude that breather formation is a robust feature. A ompelling demonstration of breather formation is the mapping of the KAM torus. We used a strobosopi method of viewing this torus whih allowed the further observation that there were only two nonzero radii (or ation variables) for the torus. Systemati physial observations suggest that alkali halides for whih the anion-ation mass ratio is lose to one have little or no anomaly in their luminesene deay. We found here that breather formation exhibits a similar effet: as this ratio approahes unity the breather gets more and more deloalized, and for appropriate parameters disappears ompletely. Aknowledgements The support of the projet KONTAKT MSMT P5ME73 and of NSF grant 5 5533 are gratefully aknowledged. Referenes [] V. Fleurov, Chaos 3, 676 (3). [] K. Polák, M. Nikl, and E. Mihóková, J. Lumin. 54, 89 (99). [3] B. Gaveau et al., Phys. Rev. B 58, 6938 (998). [4] B. Gaveau et al., J. Lumin. 9, 3 (). [5] E. Mihóková et al., Phys. Rev. B 66, 55 (). [6] L. S. Shulman et al., Phys. Rev. Lett. 88, 4 (). [7] S. A. Kiselev et al., Phys. Rev. B 5, 935 (994). [8] L. S. Shulman, Phys. Rev. A 68, 59 (3). 6 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim