Stochastic and deterministic domino processes in photoinduced structural changes

Journal of Physics and Chemistry of Solids 60 (1999) 1915–1919 www.elsevier.nl/locate/jpcs

Stochastic and deterministic domino processes in photoinduced structural changes K. Koshino a,b,*, T. Ogawa a,c a

Department of Physics, Tohoku University, Aoba-ku, Sendai 980-8578, Japan Institute of Physics, University of Tokyo, Meguro-ku, Tokyo 153-8902, Japan c Structure and Transformation, PRESTO, Japan Science and Technology Corporation (JST), Tokyo, Japan b

Received 26 April 1999; accepted 20 May 1999

Abstract The photoinduced structural change is discussed theoretically using a one-dimensional model, which is composed of localized electrons and lattices. We clarify the condition for the adiabatic and/or diabatic approximations. The global structural change by one-site excitation is possible in both cases. Under the adiabatic picture, the domain wall between the two phases moves at constant velocity, and the structural change in each site occurs nonradiatively within the time order of the lattice vibration. On the contrary, under the diabatic picture, the motion of the domain wall becomes a stochastic process because the structural change in each site occurs via the radiative transition, which takes a longer time than the lattice vibration. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Electron–lattice system; D. Phase transitions; D. Lattice dynamics

1. Introduction A system that shows the cooperative phenomena such as phase transition triggered by external light stimulation is an attractive target for material science [1]. Actually, in some organic materials such as p-conjugated polymers [2], charge transfer crystals [3] and spin-crossover complexes [4], the switching between absolutely-stable and metastable structures can be induced by local irradiation of light. The initial stage of such phenomena is closely related to the homogeneous nucleation [5] triggered by the photoexcitation of an electron, which leads to global (first-order) structural phase transitions in a nonequilibrium situation. As compared to the equilibrium phase transitions, nonequilibrium phase changes are much less understood due to the lack of a general theoretical framework [5,6]. It is, therefore, important and necessary to study simple theoretical models amenable to analysis in order to clarify the microscopic mechanisms of such transitions. * Corresponding author. Fax: 1 81-22-217-6447. E-mail address: [email protected] (K. Koshino)

In this paper, we propose a phenomenological but essential model and discuss it in both the adiabatic [7,8] and diabatic regimes. One of the purposes of this paper is to clarify qualitative differences in the spatiotemporal dynamics during the structural change process in the adiabatic and diabatic regimes. We will stress that both regimes yield two typical types of domain-growing dynamics in the photoinduced nucleation, therefore their features are universal in photoinduced global structural changes.

2. Model 2.1. Effective Hamiltonian To discuss the photoinduced cooperative structural change, a one-dimensional model was proposed, each site of which was composed of a two-level localized electron and a lattice distortion [7–10]. The state of a single site is specified by the distortion u~j of the lattice, which we treat classically, and the wavefunction uflj of the localized electron, which we assume to be a linear combination of two states, u1lj and u2lj . The Hamiltonian for this system is given

0022-3697/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00216-4

1916

K. Koshino, T. Ogawa / Journal of Physics and Chemistry of Solids 60 (1999) 1915–1919

Hint ˆ

X

kij …ui 2 uj †2 ;

…7†

i.j

with

Fig. 1. Diabatic potentials, ed1;2 , and adiabatic potentials, ea^ .

~ j and the interaction by the sum of the individual part H ~ int : H X ~ ˆ ~ int : ~ j1H H …1† H j

~ j for the jth site is the following The Hamiltonian H operator acting on the jth electronic wavefunction, 2 d 3 e~ 1 …u~j † 1 p~2j =2M~ t~ 4 5; ~ Hj ˆ …2† t~ e~ d2 …u~j † 1 p~2j =2M~ ~ are the momentum and mass of the jth where p~j and M lattice, t~ is the coupling between the two electronic states, which is independent of u~j under the Condon approximation, and e~ d1;2 …u~j † represents the diabatic potentials for the jth lattice:

e~ d1 …u~j † ˆ k~0 u~ 2j =2 2 g~ u~j 1 e~ ;

…3†

e~ d2 …u~j † ˆ k~0 u~ 2j =2 1 g~ u~j ;

…4†

where k~0 is the elastic constant, g~ the electron–lattice coupling, and e~ the energy difference between the potential minima. We assume that the intersite interaction is brought about by the lattice–lattice coupling of the following form, X ~ int ˆ H …5† k~ij …u~i 2 u~j †2 ; i.j

which acts to decrease the differences in the distortions. In the above expression, k~ij should be regarded as the renormalized value representing all other types of interactions. Using the characteristic energy e~ c ˆ g~ 2 =2k~0 , length ~ k~0 †1=2 ; and frequency u~c ˆ g~ =k~0 , momentum p~ c ˆ g~ …M= 2 1=2 V~ c ˆ …2e~Pc =M~ u~c † ; we can rewrite the Hamiltonian H ; ~ e~ c ˆ j Hj 1 Hint in the following dimensionless H= form: 2 d 3 e1 …uj † 1 p2j t 5; Hj ˆ 4 …6† t ed2 …uj † 1 p2j

ed1 …uj † ˆ u2j 2 2uj 1 e;

…8†

ed2 …uj † ˆ u2j 1 2uj ;

…9†

where uj ˆ u~j =u~c , pj ˆ p~j =p~c , e ˆ e~ =e~ c , t ˆ t~=e~ c , kij ˆ k~ij =k~0 . The time variable is denoted by the dimensionless variable t, which is normalized by V~ 21 c . As a dynamical process we consider the motion of the lattice, and the absorption and emission of the photons according to the Franck–Condon principle. Though not explicitly shown in the Hamiltonian, the lattices are coupled to the reservoir, composed of other phonon modes in the system. Thus, the system is an open one, and therefore, the total energy dissipates during the lattice relaxation. We note that our aim here is not to study this model in full generality, but rather to identify the mechanism of a photoinduced phase transition in the simplest possible scenario involving the microscopic processes. Indeed, we will show below that even within this simplest model, novel and universal photoinduced structural phase transitions occur which are nontrivial, yet amenable to analysis. 2.2. Adiabatic and diabatic regimes To observe preliminarily the properties attatched to a single site, we first neglect the intersite interaction, Hint , and investigate Hj . As shown in Fig. 1, two stable structures are allowed [7] under an appropriate choice of the parameters, e and t; in the A(B) structure, the electronic state is almost u2lj …u1lj † and the distortion uj ˆ uA . 21 …uj ˆ uB . 1†. In the following, we treat the case of e , 0, i.e. the A(B) structure is the metastable (stable) structure. It is well known that the lattices should be regarded to move in the diabatic potentials ed1;2 …uj †, if they move fast, while they should be regarded to move in the adiabatic potentials ea^ …uj †, if they move slowly. The adiabatic potentials are defined as the eigenvalues of the single-site Hamiltonian such as s  2 e e a 2 …10† e^ …uj † ˆ uj 1 ^ 2 2uj 1t2 : 2 2 To express the conditions in order to apply the above two pictures, we employ the following dimensionless constant S here, S ˆ e~ c =É~ V~ c ;

…11†

which is the so-called Huang–Rhys factor. The condition to apply the diabatic (adiabatic) picture is expressed by t p S21=2 …t q S21=2 †. We restrict our investigation to these two limiting cases in this paper.

K. Koshino, T. Ogawa / Journal of Physics and Chemistry of Solids 60 (1999) 1915–1919

1917

k and the force range m as kij ˆ

! k ui 2 ju 2 1 : ‰…1 2 exp…21=m†Šexp 2 2 m

…12†

In the limit m ! 0, it approaches the nearest neighbor interaction kij ˆ …k=2†dui2ju;1 , while in the limit m ! ∞, it approaches the infinite ranged constant kij ˆ k=N, where N is the total number of sites. 3.1. In the adiabatic regime We shall discuss the problem under the adiabatic repesentation [7,9,10]. When one of the sites in this system (the zeroth site) is excited, the lattices start to relax from the initial position uA in the one-site-excited adiabatic potential Ea1 , which is given by X a X e2 …uj † 1 ea1 …u0 † 1 kij …ui 2 uj †2 ; …13† Ea1 …{uj }† ˆ i.j

j…±0†

to the relaxed lattice configuration {ua1 j }. This is a local distortion centered about the zeroth site, and this relaxation finishes within the time order of the lattice vibration, i.e. tvib < 100–1 . After this relaxation, the zeroth site is still unstable against the radiative decay, and in the time order of the spontaneous emission (the spontaneous emission lifetime tSE < 104–5 ), it decays to the ground state. The lattices then start to relax again from {u a1 j } in the ground-state adiabatic potential Ea0 , which is given by X a X Ea0 …{uj }† ˆ e2 …uj † 1 kij …ui 2 uj †2 : …14† j

d1 Fig. 2. Schematic view of the relaxed positions, u d1 0 and u 1 , just d1 d  after photoexcitation: (a) ud1 , u , u (phase I ), (b) u , ud1 X X 1 0 1 , d1 d1 d1 d d u0 (phase II ), (c) u1 , u0 , uX (phase III ).

3. Structural change after one-site excitation Now we start investigating the structural change from the metastable A structure (uj ˆ uA for all j), considering the intersite interaction Hint . As the simplest case of the photoinduced dynamics, we focus on the case that a single photon is absorbed at a certain site, which we call the zeroth site. As our main interest lies in the photoinduced dynamics, we consider only the case that the lattice temperature is absolute zero in order to exclude the effect of thermal fluctuation. For simplicity, we hereafter assume that the intersite interaction can be parameterized by the total strength

i.j

At this stage, we can observe the following three qualitatively different cases: (a) only the excited site makes a structural change to B (phase I a), (b) every site yields a structural change to B (phase II a), and (c) every site goes back to the A structure (phase III a). In phase II a, the region of the B structure extends step by step with a constant velocity (called the deterministic domino effect), and each site changes its structure nonradiatively within the time order of the lattice vibration [7,8]. 3.2. In the diabatic regime Under the diabatic picture, on the contrary, the system relaxes in a qualitatively different way, which we discuss in detail in the following. When the zeroth site is excited, the system starts to relax in the one-site-converted diabatic potential Ed1 , which is given by X d X Ed1 …{uj }† ˆ e2 …uj † 1 ed1 …u0 † 1 kij …ui 2 uj †2 ; …15† j…±0†

i.j

to the relaxed lattice configuration {ud1 j }, which is given by 2 Zp cos…qj† u d1 dq ; …16† j ˆ 21 1 p 0 v2q

1918

K. Koshino, T. Ogawa / Journal of Physics and Chemistry of Solids 60 (1999) 1915–1919

and returns to the initial A structure. We call this case phase III d. (iii) Finally, we discuss the case of uX , ud1 j not only for j ˆ 0 but also for j ˆ ^1. In this case, the ^1st sites are in the higher electronic state, and have the possibility of radiative decay. As the spontaneous emission from these sites is a stochastic process, we cannot predict which site emits a photon earlier. For example, we treat the case that the 1st site emits a photon before the 21st site does. Then the lattices start relaxation in the two-site-converted diabatic potential Ed2 given by X X X Ed2 …{uj }† ˆ e1 …uj † 1 e2 …uj † 1 kij …ui 2 uj †2 jˆ0;1

i.j

j±0;1

…19† to the new relaxed configuration ud2 j ˆ 21 1

{ud2 j },

which is given by

X 2 Zp cos‰q…j 2 l†Š dq : p 0 v2q lˆ0;1

…20†

Fig. 3. The phase diagram under the diabatic picture (t ˆ 0) on (m,k) plane with (a) e ˆ 23 and (b) e ˆ 22.

In this new configulation, the 2nd and (21)st sites become unstable against the radiative decay. After all, the region of the B structure extends through such stochastic sequence of the radiative decays (called the stochastic domino effect), which we call phase II d.

where

3.3. Phase diagram in the diabatic regime

v2q

ˆ11k2

X

k0j exp…2iqj†:

…17†

j…±0†

This is a local lattice distortion centered about the zeroth site, which is similar to {u a1 j } in some sense. After this relaxation, the system may still have the instability against the radiative decay. The stability against the radiative decay is determined by the relation between the crossing point uX …ˆ e=4† of the two diabatic potentials, and the relaxed position ud1 j of the lattice. (i) First, we discuss the case of uX , ud1  d1 0 and u j , uX for j ± 0, which occurs when the intersite interaction is weak (small k). As shown in Fig. 2(a), every site is in the lower electronic state, and stable against the radiative decay. This local distortion does neither grow nor disappear, which we hereafter call phase I d. (ii) Next, we discuss the case of ud1 j , uX for all j, which occurs when the intersite interaction is strong (large k). As shown in Fig. 2(c), the zeroth site is in the higher electronic state and unstable against the radiative decay. It will decay radiatively to the other electronic states in the time order of tSE . After emission of the photon, the system starts relaxation again in the zero-site-converted diabatic potential Ed0 , which is given by X X Ed0 …{uj }† ˆ e2 …uj † 1 kij …ui 2 uj †2 ; …18† j

i.j

As mentioned above, the one-site excitation in A structure results in three qualitatively different behavior. This is determined by the relation between uX ˆ e=4, and {ud1 j }, which is determined by k and m. In Fig. 3, the phase diagrams on the (k,m ) plane are drawn, with the parameter e fixed. The crucial difference between Fig. 3(a) and (b) is the existence of the phase II d, where a global structural change is induced. The condition for the existence of phase II d is given by p 24 , e , 4…5 2 4 2†; …21† which indicates the fragility of the metastable A structure. The lower limit corresponds to the condition of bistability of the system. Under condition (21), the phase II d appears in the region of small m, as shown in Fig. 3(a). The parameters in the figure are given by kId –IIId …0† ˆ

1 22 …a 2 1†; 2

kId –IIId …∞† ˆ a21 2 1;

kI^d –IId …0†

…22† …23†

p 2…a2 1 4a 1 1† ^ …a 2 1†2 …a2 2 6a 1 1† ˆ ; 4a2 …24†

K. Koshino, T. Ogawa / Journal of Physics and Chemistry of Solids 60 (1999) 1915–1919

1919

Table 1 Comparison between the domino dynamics under the diabatic and adiabatic pictures after excitation of the zeroth site

Condition Phase Spontaneous emission at j ˆ 0 Spontaneous emission at j ± 0 Growth of the domain wall Time scale of the domain-wall motion

Diabatic picture

Adiabatic picture

t p S21=2 II d None Occurs Stochastic tSE < 104–5

t q S21=2 II a Occurs None Deterministic tvib < 100–1

where

e 1 aˆ 1 : 2 8

…25†

The results that phase II d appears only when the metastable structure is fragile, and that it appears in the region of small m are qualitatively same as the results given under the adiabatic picture [8]. 4. Summary and discussion In summary, the possibility of the global structural change by one-site excitation (phases II d and II a) is confirmed under both the diabatic and adiabatic pictures. The physical processes during the structural change, however, are quite different from each other, which are summarized in Table 1. Under the adiabatic picture, the global structural change is characterized as the adiabatic motion of the lattices in the ground-state adiabatic potential Ea0 given by Eq. (14), triggered by the local distortion {ua1 j } injected by the photoexcitation. The structural change in each site occurs nonradiatively along the adiabatic potential ea2 , and finishes within the time order of the lattice vibration. The domain wall, between the A and B structures, moves deterministically at a constant velocity. On the contrary, under the diabatic picture, the global structural change is characterized as a sequence of radiative decays followed by lattice relaxations. The structural change in each site accompanies a spontaneous emission, and takes a much longer time than the nonradiative structural change. The motion of the domain wall now becomes a stochastic process due to the spontaneous emission in each site. We have to mention the weak points of our model. In our model, the lattice dynamics is treated in a classical way. This is justified only in the adiabatic and diabatic limits. In the crossover regime, where the nonadiabatic processes take place, the lattice dynamics is no longer described by the classical equations of motion, but by their quantum and stochastic natures. In such a case, domino process becomes

intermediate between the deterministic and stochastic ones, thus the total number of emitted photons becomes less than N 2 1. Then the domain walls move intermittently. This situation will be reported elsewhere [11]. We also have to study the case of multisite photoexcitation, corresponding to experimental situations. When adjacent sites are simultaneously excited by light, such a “domain” with a finite length can develop to global size even in the phase I a or I d. This indicates the existence of a critical droplet size of nucleation in one dimension, i.e. the threshold-like behavior. Acknowledgements The authors are grateful to S. Koshihara, T. Luty, and H. Cailleau for stimulating discussion. This work is partly supported by the Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan.

References [1] K. Nasu (Ed.), Relaxations of Excited States and PhotoInduced Structural Phase Transitions Springer, Berlin, 1997. [2] S. Koshihara, Y. Tokura, K. Takeda, T. Koda, Phys. Rev. Lett. 68 (1992) 1148. [3] S. Koshihara, Y. Tokura, T. Mitani, G. Saito, T. Koda, Phys. Rev. B 42 (1990) 6853. [4] A. Hauser, A. Vef, P. Adler, J. Chem. Phys. 95 (1991) 8710. [5] J.D. Gunton, M.S. Miguel, P.S. Sahni, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, 8, Academic Press, London, 1983. [6] K. Kawasaki, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, 2, Academic Press, London, 1972. [7] K. Koshino, T. Ogawa, J. Phys. Soc. Jpn 67 (1998) 2174. [8] K. Koshino, T. Ogawa, Phys. Rev. B 58 (1998) 14804. [9] E. Hanamura, N. Nagaosa, J. Phys. Soc. Jpn 56 (1987) 2080. [10] N. Nagaosa, T. Ogawa, Phys. Rev. B 39 (1989) 4472. [11] K. Koshino, T. Ogawa, J. Korean Phys. Soc. 34 (1999) S21.