- Email: [email protected]
Elementary nonlinear processes in the formation of domains after photoexcitation
Journal of Luminescence 87}89 (2000) 655}657
Elementary nonlinear processes in the formation of domains after photoexcitation Hideo Mizouchi* Institute of Materials Structure Science, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract We theoretically study lattice relaxation dynamics of a photogenerated exciton which "nally results in a macroscopic domain in insulators. Especially, we focus on three elementary nonlinear processes of this domain formation; aggregation, growth and proliferation. We consider a strongly coupled many exciton-Einstein phonon system. We also assume the system to interact with a reservoir, in order to take into account the relaxation channels such as vibrational relaxations, non-radiative and radiative decays of excitons. The excitons are assumed to be strongly repulsive on the same sites and attractive on the nearest-neighboring sites. The third-order anharmonic coupling between excitons is also assumed. Our system is in the so-called multistable situation where various multi-exciton states are energetically close to 1-exciton states. Within the Markov approximation for the reservoir, we investigate full-quantummechanically the time evolution of the system. We have shown that in our model, the aforementioned elementary processes successfully work against the relaxation channels. We conclude the anharmonicity and the multistability of excitons are indispensable for the domain formation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Photoinduced structural phase transition; Elementary nonlinear processes; Exciton}phonon system; Reservoir
1. Introduction As is well-known, when a visible light is shone on an insulator, an electron is excited, and then it induces a change of the lattice structure. For a long time, this change has been believed to occur only around a small number of atoms. However, the following interesting phenomena are recently discovered in some insulators. In such insulators, the microscopic structural change grows up to be a macroscopic domain with new electronic and lattice orders. These phenomena are called &photoinduced structural phase transition' (PISPT). For example, in an organic charge transfer salt, TTF-CA, one photon creates about 200 neutral TTF-CA pairs in its ionic ground state [1]. In order to create such a macroscopic domain, the following elementary nonlinear processes are considered to be important. (1) Two nucleation kernels created simultaneously by photons, aggregate each
other during the relaxation processes and "nally result in a macroscopic domain (Aggregation). (2) When a new kernel merges into an old domain existing already, the resultant yield of the domain formation is larger than the case (1) (Growth). (3) A kernel created initially by a photon proliferates afterwards (Proliferation). In this paper, we focus on these three elementary processes and study the early time dynamics of PISPT. In such early stage, nucleation kernels do not stay within only one adiabatic potential surface but transit diabatically between the di!erent potential ones. Furthermore, various dissipation channels are also signi"cant. Therefore, we treat lattice vibration quantummechanically, and a reservoir is considered in order to treat dissipation and relaxation processes microscopically, as shown in the next section.
2. Model Hamiltonian * Fax: #81-298-64-2801. E-mail address: [email protected] (H. Mizouchi)
We consider the one-dimensional exciton}phonon system interacting with a reservoir. The total Hamiltonian
0022-2313/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 3 4 5 - 2
656
H. Mizouchi / Journal of Luminescence 87}89 (2000) 655}657
H is as follows H"H #H #H .
(1)
In Eq. (1), H denotes the Hamiltonian of the exciton}phonon system. This is our relevant system, and is written as H " t("l!l ")BRB J JY J JY # f ("l!l ")BRB BR B J J JY JY J JY # g("l!l ")BRB (BR #B ) J J JY JY J JY #u bRb J J J !u(S BRB (bR#b ). J J J J J
(2)
Here, B and b are annihilation operators of an exciton J J and an Einstein phonon, respectively, at a site l. The single and double prime sums denote the sums under the condition that l*l and lOl , respectively. In Eq. (2), the "rst and second terms represent one-body and twobody interactions of excitons, respectively. In the second term, we assume the on-site repulsion and the intersite attraction. The third term represents the third-order anharmonic coupling between excitons, and through this coupling, an exciton is created and annihilated by another one. These interactions among excitons result naturally from the original Coulomb interactions among electrons constituting excitons. The fourth term represents the energy of Einstein phonons with a frequency, u. The last term represents the on-site exciton}phonon interaction with a dimensionless coupling constant, S. In Eq. (1), H denotes the Hamiltonian of the reservoir, which is composed of the even and odd bosonic modes and photons. H denotes the interactions between the system and the reservoir. Here, the even modes couple linearly with phonons and dissipate them, while the odd modes (photons) couple linearly with excitons and nonradiatively (radiatively) create and annihilate them. Therefore, through these interactions, we can take into account the relaxation channels such as vibrational (or phonon) relaxations, non-radiative and radiative transitions of excitons, respectively. The parameter values are taken as follows. The size of the exciton}phonon system is taken as four sites. The periodic boundary condition is imposed on the system. As for the interactions of excitons in the system, we consider only the creation energy t(0), the nearest-neighbor hopping energy t(1), the on-site interaction f (0)('0), the intersite interaction f (1)((0) and the anharmonic coupling g(1). Here, we take f (0)"#R, neglecting the
states with more than two excitons at the same site. In the following, we use <(,!f (1)) and g(,g(1)). We take u"0.1 eV, S"3 or 8, t(0)/u"9.5, t(1)/u"!1.5,
3. Results and discussion In the following, q denotes the time measured after photoexcitation has been completed, and ¹ denotes one phonon period, 2n/u. The ratios of 0-, 1-, 2- and 3exciton states are represented by rat0, rat1, rat2, and rat3, respectively. The density}density correlation functions between nearest-neighboring excitons, and between second nearest-neighboring excitons are represented by dd1 and dd2, respectively. At "rst, we show the time evolution behavior of the system when the exciton}phonon coupling constant S is changed. Here, we consider from 0- to 3-exciton states. The initial state is taken as the Franck}Condon state with only one exciton localized at a site. In Fig. 1a, the time evolution of the energy in the system is shown. When S"8, the energy decreases rapidly from the initial one. This is mainly due to vibrational relaxations. As time goes by, the decrease gets slower and slower, and the system stays in the photoexcited states for a long time. When S"3, the energy decreases from the initial one, and then, the decrease gets a little slower. However, unlike the large S case, the decrease continues apparently. From Fig. 1b, the total number of excitons increases at an early time when S"8, while it continues to decrease from 1 when S"3. This shows that in the small S case, the exciton continues to annihilate with radiative and non-radiative transitions, and the system goes back to the ground state before forming any domains. Therefore,
H. Mizouchi / Journal of Luminescence 87}89 (2000) 655}657
Fig. 1. The time evolution of the energy in the system (a), and of the total number of excitons (b). The initial state is taken to be the Franck}Condon state with one exciton localized at a site. u and ¹ denote the phonon frequency and its period, respectively.
S must be large enough in order to obtain stable domains. In the following, S is "xed to be equal to 8. Next, we show the results of the aggregation process. Here, we consider from 0- to 2-exciton states. The initial state is the Franck}Condon state with two excitons sitting at second nearest-neighbor sites to each other. At "rst, the energy decreases rapidly from the initial one, and then, the decrease gets slower and slower. rat2 decreases rapidly from 1, while rat1 increases rapidly from 0. When q*2 T, the change of them is suppressed, and then, they decreases very slowly. When q"7 T, rat1 and rat2 are 0.60 and 0.37, respectively. rat0 continues to increase gradually from 0. In the initial state, dd1(dd2) is equal to 0(0.5). Just after the initial time, dd1 increases and dd2 decreases. When q*7 T, dd1 decreases very slowly, and keeps the positive value for a long time. This shows that the two excitons, which initially sit away from each other, come and keep close to each other. This is the aggregation. When < is small, rat2 and dd1 do not remain for a long time. In this case, the system is not in the multistable situation, and multi-exciton states relax easily to the low-energy states with the smaller number of excitons. Therefore, the multistability of exciton states is necessary for the aggregation. The results of the process of the growth are as follows. We consider from 0- to 2-exciton states. In the previous case of the aggregation, the two excitons are photogenerated simultaneously, while, in the process of the growth, two excitons are photogenerated one by one at a interval of 10 T. The initial time of the growth process is de"ned as the time when the second exciton is photogenerated. At "rst, the initial energy is lower than the case of the aggregation because the "rst exciton is already relaxed in the initial state. When q*5 T, rat2 and dd1
657
Fig. 2. The time evolution of the ratios of 0-, 1-, 2- and 3-exciton states (a), and of the density}density correlation functions of excitons (b). S"8 and the initial state is the same as the case in Fig. 1. In (a), rat0, rat1, rat2, and rat3 denote the ratios of 0-, 1-, 2- and 3-exciton states, respectively. In (b), dd1 and dd2 denote the correlation functions between nearest-neighboring excitons, and between second nearest-neighboring ones, respectively.
are larger than the case of the aggregation by 0.03 and 0.003, respectively. This is the growth. This is because the relaxed "rst exciton hardly decays, and thus, the decay rate of the 2-exciton states is also low. Finally, we show the process of the proliferation. Here, we consider from 0- to 3-exciton states, and the initial state is the same as the case in Fig. 1. The time evolution behavior of the energy and the total number of excitons are already shown in Fig. 1. In Fig. 2a, rat1 decreases rapidly from 1, while rat2 and rat3 increase rapidly from 0. When q*5 T, the change of them is suppressed. At q"5 T, rat1, rat2 and rat3 are equal to 0.47, 0.28 and 0.22, respectively. This is the proliferation. Furthermore, from Fig. 2b, dd1 and dd2 increase rapidly from 0 and then keep the positive values for a long time. This shows that the excitons not only proliferate but also aggregate each other. When g is small, rat2 and rat3 cannot be large. When < is small, they do not remain for a long time. Therefore, the anharmonicity and the multistability of exciton states are necessary for the proliferation. We have shown that in our model, the three elementary processes of PISPT successfully work against dissipation and relaxation channels. We conclude the anharmonicity and the multistability of excitons are indispensable for the occurrence of PISPT.
References [1] S. Koshihara, Y. Takahashi, H. Sakai, Y. Tokura, T. Luty, J. Phys. Chem. B 103 (1999) 2592. [2] K. Cho, Y. Toyozawa, J. Phys. Soc. Jpn. 30 (1971) 1555.
Elementary nonlinear processes in the formation of domains after photoexcitation Hideo Mizouchi* Institute of Materials Structure Science, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract We theoretically study lattice relaxation dynamics of a photogenerated exciton which "nally results in a macroscopic domain in insulators. Especially, we focus on three elementary nonlinear processes of this domain formation; aggregation, growth and proliferation. We consider a strongly coupled many exciton-Einstein phonon system. We also assume the system to interact with a reservoir, in order to take into account the relaxation channels such as vibrational relaxations, non-radiative and radiative decays of excitons. The excitons are assumed to be strongly repulsive on the same sites and attractive on the nearest-neighboring sites. The third-order anharmonic coupling between excitons is also assumed. Our system is in the so-called multistable situation where various multi-exciton states are energetically close to 1-exciton states. Within the Markov approximation for the reservoir, we investigate full-quantummechanically the time evolution of the system. We have shown that in our model, the aforementioned elementary processes successfully work against the relaxation channels. We conclude the anharmonicity and the multistability of excitons are indispensable for the domain formation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Photoinduced structural phase transition; Elementary nonlinear processes; Exciton}phonon system; Reservoir
1. Introduction As is well-known, when a visible light is shone on an insulator, an electron is excited, and then it induces a change of the lattice structure. For a long time, this change has been believed to occur only around a small number of atoms. However, the following interesting phenomena are recently discovered in some insulators. In such insulators, the microscopic structural change grows up to be a macroscopic domain with new electronic and lattice orders. These phenomena are called &photoinduced structural phase transition' (PISPT). For example, in an organic charge transfer salt, TTF-CA, one photon creates about 200 neutral TTF-CA pairs in its ionic ground state [1]. In order to create such a macroscopic domain, the following elementary nonlinear processes are considered to be important. (1) Two nucleation kernels created simultaneously by photons, aggregate each
other during the relaxation processes and "nally result in a macroscopic domain (Aggregation). (2) When a new kernel merges into an old domain existing already, the resultant yield of the domain formation is larger than the case (1) (Growth). (3) A kernel created initially by a photon proliferates afterwards (Proliferation). In this paper, we focus on these three elementary processes and study the early time dynamics of PISPT. In such early stage, nucleation kernels do not stay within only one adiabatic potential surface but transit diabatically between the di!erent potential ones. Furthermore, various dissipation channels are also signi"cant. Therefore, we treat lattice vibration quantummechanically, and a reservoir is considered in order to treat dissipation and relaxation processes microscopically, as shown in the next section.
2. Model Hamiltonian * Fax: #81-298-64-2801. E-mail address: [email protected] (H. Mizouchi)
We consider the one-dimensional exciton}phonon system interacting with a reservoir. The total Hamiltonian
0022-2313/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 3 4 5 - 2
656
H. Mizouchi / Journal of Luminescence 87}89 (2000) 655}657
H is as follows H"H #H #H .
(1)
In Eq. (1), H denotes the Hamiltonian of the exciton}phonon system. This is our relevant system, and is written as H " t("l!l ")BRB J JY J JY # f ("l!l ")BRB BR B J J JY JY J JY # g("l!l ")BRB (BR #B ) J J JY JY J JY #u bRb J J J !u(S BRB (bR#b ). J J J J J
(2)
Here, B and b are annihilation operators of an exciton J J and an Einstein phonon, respectively, at a site l. The single and double prime sums denote the sums under the condition that l*l and lOl , respectively. In Eq. (2), the "rst and second terms represent one-body and twobody interactions of excitons, respectively. In the second term, we assume the on-site repulsion and the intersite attraction. The third term represents the third-order anharmonic coupling between excitons, and through this coupling, an exciton is created and annihilated by another one. These interactions among excitons result naturally from the original Coulomb interactions among electrons constituting excitons. The fourth term represents the energy of Einstein phonons with a frequency, u. The last term represents the on-site exciton}phonon interaction with a dimensionless coupling constant, S. In Eq. (1), H denotes the Hamiltonian of the reservoir, which is composed of the even and odd bosonic modes and photons. H denotes the interactions between the system and the reservoir. Here, the even modes couple linearly with phonons and dissipate them, while the odd modes (photons) couple linearly with excitons and nonradiatively (radiatively) create and annihilate them. Therefore, through these interactions, we can take into account the relaxation channels such as vibrational (or phonon) relaxations, non-radiative and radiative transitions of excitons, respectively. The parameter values are taken as follows. The size of the exciton}phonon system is taken as four sites. The periodic boundary condition is imposed on the system. As for the interactions of excitons in the system, we consider only the creation energy t(0), the nearest-neighbor hopping energy t(1), the on-site interaction f (0)('0), the intersite interaction f (1)((0) and the anharmonic coupling g(1). Here, we take f (0)"#R, neglecting the
states with more than two excitons at the same site. In the following, we use <(,!f (1)) and g(,g(1)). We take u"0.1 eV, S"3 or 8, t(0)/u"9.5, t(1)/u"!1.5,
3. Results and discussion In the following, q denotes the time measured after photoexcitation has been completed, and ¹ denotes one phonon period, 2n/u. The ratios of 0-, 1-, 2- and 3exciton states are represented by rat0, rat1, rat2, and rat3, respectively. The density}density correlation functions between nearest-neighboring excitons, and between second nearest-neighboring excitons are represented by dd1 and dd2, respectively. At "rst, we show the time evolution behavior of the system when the exciton}phonon coupling constant S is changed. Here, we consider from 0- to 3-exciton states. The initial state is taken as the Franck}Condon state with only one exciton localized at a site. In Fig. 1a, the time evolution of the energy in the system is shown. When S"8, the energy decreases rapidly from the initial one. This is mainly due to vibrational relaxations. As time goes by, the decrease gets slower and slower, and the system stays in the photoexcited states for a long time. When S"3, the energy decreases from the initial one, and then, the decrease gets a little slower. However, unlike the large S case, the decrease continues apparently. From Fig. 1b, the total number of excitons increases at an early time when S"8, while it continues to decrease from 1 when S"3. This shows that in the small S case, the exciton continues to annihilate with radiative and non-radiative transitions, and the system goes back to the ground state before forming any domains. Therefore,
H. Mizouchi / Journal of Luminescence 87}89 (2000) 655}657
Fig. 1. The time evolution of the energy in the system (a), and of the total number of excitons (b). The initial state is taken to be the Franck}Condon state with one exciton localized at a site. u and ¹ denote the phonon frequency and its period, respectively.
S must be large enough in order to obtain stable domains. In the following, S is "xed to be equal to 8. Next, we show the results of the aggregation process. Here, we consider from 0- to 2-exciton states. The initial state is the Franck}Condon state with two excitons sitting at second nearest-neighbor sites to each other. At "rst, the energy decreases rapidly from the initial one, and then, the decrease gets slower and slower. rat2 decreases rapidly from 1, while rat1 increases rapidly from 0. When q*2 T, the change of them is suppressed, and then, they decreases very slowly. When q"7 T, rat1 and rat2 are 0.60 and 0.37, respectively. rat0 continues to increase gradually from 0. In the initial state, dd1(dd2) is equal to 0(0.5). Just after the initial time, dd1 increases and dd2 decreases. When q*7 T, dd1 decreases very slowly, and keeps the positive value for a long time. This shows that the two excitons, which initially sit away from each other, come and keep close to each other. This is the aggregation. When < is small, rat2 and dd1 do not remain for a long time. In this case, the system is not in the multistable situation, and multi-exciton states relax easily to the low-energy states with the smaller number of excitons. Therefore, the multistability of exciton states is necessary for the aggregation. The results of the process of the growth are as follows. We consider from 0- to 2-exciton states. In the previous case of the aggregation, the two excitons are photogenerated simultaneously, while, in the process of the growth, two excitons are photogenerated one by one at a interval of 10 T. The initial time of the growth process is de"ned as the time when the second exciton is photogenerated. At "rst, the initial energy is lower than the case of the aggregation because the "rst exciton is already relaxed in the initial state. When q*5 T, rat2 and dd1
657
Fig. 2. The time evolution of the ratios of 0-, 1-, 2- and 3-exciton states (a), and of the density}density correlation functions of excitons (b). S"8 and the initial state is the same as the case in Fig. 1. In (a), rat0, rat1, rat2, and rat3 denote the ratios of 0-, 1-, 2- and 3-exciton states, respectively. In (b), dd1 and dd2 denote the correlation functions between nearest-neighboring excitons, and between second nearest-neighboring ones, respectively.
are larger than the case of the aggregation by 0.03 and 0.003, respectively. This is the growth. This is because the relaxed "rst exciton hardly decays, and thus, the decay rate of the 2-exciton states is also low. Finally, we show the process of the proliferation. Here, we consider from 0- to 3-exciton states, and the initial state is the same as the case in Fig. 1. The time evolution behavior of the energy and the total number of excitons are already shown in Fig. 1. In Fig. 2a, rat1 decreases rapidly from 1, while rat2 and rat3 increase rapidly from 0. When q*5 T, the change of them is suppressed. At q"5 T, rat1, rat2 and rat3 are equal to 0.47, 0.28 and 0.22, respectively. This is the proliferation. Furthermore, from Fig. 2b, dd1 and dd2 increase rapidly from 0 and then keep the positive values for a long time. This shows that the excitons not only proliferate but also aggregate each other. When g is small, rat2 and rat3 cannot be large. When < is small, they do not remain for a long time. Therefore, the anharmonicity and the multistability of exciton states are necessary for the proliferation. We have shown that in our model, the three elementary processes of PISPT successfully work against dissipation and relaxation channels. We conclude the anharmonicity and the multistability of excitons are indispensable for the occurrence of PISPT.
References [1] S. Koshihara, Y. Takahashi, H. Sakai, Y. Tokura, T. Luty, J. Phys. Chem. B 103 (1999) 2592. [2] K. Cho, Y. Toyozawa, J. Phys. Soc. Jpn. 30 (1971) 1555.