PRL 110, 116401 (2013)
week ending 15 MARCH 2013
PHYSICAL REVIEW LETTERS
Revealing the Photorelaxation Mechanism in a Molecular Solid Using Density-Functional Theory K. Iwano1 and Y. Shimoi2 1
Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), Graduate University for Advanced Studies, 1-1 Oho, Tsukuba 305-0801, Japan 2 Nanosystem Research Institute (NRI), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba 305-8568, Japan (Received 1 October 2012; revised manuscript received 12 January 2013; published 11 March 2013) Photorelaxation in a molecular crystal is investigated by a density-functional theory for the first time. A quasi-one-dimensional molecular compound, ðEDO-TTFÞ2 PF6 , is known to exhibit a photoinduced phase transition, which is characterized as a transition from a (0110)-type charge-ordering insulator phase to a high-temperature metallic phase. First, we apply the method of embedding a cluster in a self-consistent environment and succeed in constructing a stable tetramer structure of EDO-TTF molecules. The reliance of this cluster is ensured by a vibrational analysis that well reproduces the IR and Raman frequencies particularly for C¼C stretching modes including a relatively large degree of electron–‘molecular vibration’ coupling. Second, relaxations in the photoexcited states of this cluster are investigated by adiabatic potential-surface analyses and full structural optimization. A reaction coordinate is found to be quite unique for a relatively high-energy excitation, namely, the so-called CT2 excitation, which is interpreted as leading to the photoinduced phase transition. DOI: 10.1103/PhysRevLett.110.116401
PACS numbers: 71.15.Mb, 71.30.+h, 78.20.Bh
The physical and chemical aspects of photoinduced phase transitions (PIPTs) have been studied extensively from both the theoretical and experimental sides. In particular, one of the interests that have been continuing since the beginning of this subject is the visualization of their dynamical processes at the atomic level, namely, the making of ‘‘molecular movies’’ [1]. This viewpoint is regarded as particularly meaningful for compounds with complicated structures, for example, molecular solids, which have both intra- and intermolecular structures and are expected to exhibit a time evolution of the correlated degrees of freedom. Such a visualization is highly expected to reveal the mechanism of the PIPTs and give us new insight into the hidden nature of the materials as well as nonequilibrium processes. We already see several successful examples realized particularly by time-resolved x-ray or electron diffraction techniques in a variety of materials [2–5]. Motivated by such rapid developments in the experiments, we aim at theoretically visualizing photorelaxations in a molecular solid most realistically, based on a densityfunctional theory (DFT), and have succeeded in identifying a special path of structural relaxation. So far, most of the studies on this topic have been based on the so-called model calculations. There are a few studies using DFTs, although they are limited to a simple system like bismuth [6–8]. The advantage of the present method is its realistic treatment of electron-nuclei systems having pure Coulombic interactions, many orbitals, and many nuclei. In the PIPTs, or photorelaxation in a more general context, the essence of the dynamics lies in optically excited states and the structural deformations associated with them. 0031-9007=13=110(11)=116401(5)
For this reason, we apply a method of a cluster to this problem, instead of that based on a periodic boundary condition, which needs the treatment of a supercell and hence a relatively high computer load. Our cluster is augmented by a self-consistent environment that mimics the atmosphere around the cluster [9,10], as will be soon explained in detail, and the computer load is then similar to those in usual cluster calculations. Our special target, ðEDO-TTFÞ2 PF6 (EDO-TTF= ethylendioxy-tetrathiafulvalene), is a quasi-one-dimensional molecular solid having two thermal equilibrium phases: a high-temperature (HT) metallic phase and a lowtemperature (LT) insulator phase [11]. In the latter phase, the EDO-TTF molecules are characterized by two types of peculiar structural features, namely, tetramerization along the one-dimensional axis and intramolecular deformations. They induce a so-called (0110)-type charge ordering (CO), with 0 and 1 being the hole occupancies of the EDO-TTF molecules in the tetramer [11,12]. The PIPT observed in ðEDO-TTFÞ2 PF6 is basically a metallization starting from this state after a considerable time of at least 100 ps after photoexcitation [13,14]. This metallization is known to be induced by exciting both the lower and higher absorption bands at 0.5 and 1.5 eV [15], known respectively as the CT1 and CT2 bands [12,16]. Our finding here is that an intermolecular mode which increases the distance between the positively charged molecules is most deeply relevant in the metallization. Furthermore, this finding is also related to a newly found (0101)-type CO or charge fluctuation claimed to appear around 0.1 ps after the photoexcitaiton [17].
116401-1
Ó 2013 American Physical Society
To investigate the photorelaxations, we have to prepare a stable molecular cluster in its electronic ground state and exclude any instability that might exist already in the ground state. We hence define our self-consistent environment as the collection of point charges [9,10] and Lennard-Jones (LJ) 6–12 type of potential sources. More specifically, as is very schematically shown in Fig. 1(d), the cluster is surrounded by two layers; the closer layer consists of both the point charges and the LJ sources, while the farther one of only the point charges. The naming of ‘‘selfconsistent’’ means that the values of the point charges are determined self-consistently so as to match those calculated quantum mechanically within the system cluster. In Figs. 1(a) and 1(b), we compare the structures optimized by our cluster approach and determined by x-ray diffraction, respectively. In Fig. 1(a), within a cluster of three tetramers, the central tetramer is allowed to move freely. Note that the anions are kept at their observed positions and structures. Regarding the details of the calculation, see the Supplementary Material [18]. As is seen in the result, we consider that the present level of the optimized structure is practically acceptable. Looking more closely, we find that some changes occur as a result of the optimization. For example, the bending angles defined in Fig. 1(c) are changed by a few degrees for the inner two molecules while they are maintained in the outer two molecules. We also emphasize that the degree of the CO is also maintained: the molecular valencies in the tetramer, (0.18, 0.82, 0.82, 0.18), for the optimized structure are very similar to those for the observed structure, (0.17, 0.83, 0.83, 0.17).
Calculated vibrational frequencies provide further evidence that ensures the stability of the present cluster model. In Fig. 2. the theoretical results as well as the experimental ones [12] in the region of C¼C stretching modes are summarized with the conventional namings of the modes in Fig. 2(c) and an explicit tetramer structure in Fig. 2(d). We consider that the overall coincidence is satisfactory at this stage of the investigation. We note that the (0, ) mode is unexpectedly calculated at a frequency higher than 1600 cm1 , because of an inaccurate mixing with a hydrogen motion. Focusing on the physical aspect itself, the lowest mode, (þ 1, ), is worth special mention. As has been argued in the preceding studies for this [12] and other materials [19], the large frequency splitting between the Ag and Au modes originates from a rather strong electron–‘molecular vibration’ coupling via electron intermolecular transfer. This is the first reproduction of this coupling based on the DFT method, and again justifies the present building of the cluster. We now proceed to the investigation of the photorelaxations. Here, we relax only the central tetramer in the three consecutive tetramers, leaving the other two tetramers in the experimentally determined configuration, as is done in the ground-state optimization described in the previous paragraphs. Excited states are described by means of the time-dependent DFT method [20]. First, we analyze adiabatic potential surfaces with respect to the following two degrees of freedom. One is the intermolecular displace˚ , and defined as in Fig. 3(a). The other ment d in units of A is the degree of intramolecular deformation s that basically
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FIG. 1 (color online). (a) and (b) Structures of three consecutive EDO-TTF tetramers optimized by our cluster approach (a) and determined by x-ray diffraction (b). (c) Definition of two bending angles. (d) Schematic picture of the two-layer model.
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FIG. 2 (color online). Calculated (a) and observed (b) frequencies of the C¼C stretching modes. The assignments mean ‘‘(valency, mode).’’ (c) Illustration of three modes, and (d) the structure and the molecular valencies in a tetramer.
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PHYSICAL REVIEW LETTERS
PRL 110, 116401 (2013)
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FIG. 3 (color online). Definitions of (a) the ‘‘d’’ mode and (b) the ‘‘s’’ mode. The stick with an open circle stands for an EDO-TTF molecule. (c) Adiabatic potential curves for the lowest thirty excited states as a function of d. (d) Potential surface for the CT2 excitation with respect to d and s.
exchanges the bent and flat molecules between s ¼ 0 and s ¼ 2, as shown in Fig. 3(b). Note that the molecular shapes in the LT and HT phases are almost reproduced at s ¼ 0 and s ¼ 1, respectively. This deformation is created using the vibrational mode of an isolated molecule [21]. In Fig. 3(c), we draw the potential curves along the d axis, with s being fixed at zero. As is seen clearly, we find substantial relaxations for the state labeled CT2. This state has the largest oscillator strength in this energy region, and its largest component is the transition localized in the central tetramer, being consistent with the present structural change confined in the same tetramer. The meaning of this relaxation is well understood from the nature of the CT2 excitation, namely, a bonding to antibonding transition [9]. Moreover, its almost isolated nature is due to the exciton character from the lowest to the highest band typically seen in the three-quarter-filled band systems. Next, we draw the potential surface of this state using both of the variables in Fig. 3(d). The minimum in the surface is located around ðd; sÞ ¼ ð0:07; 0:2Þ, suggesting that the relaxation along the intramolecular deformation is not very effective. Regarding the other excitations, we do not recognize any substantial relaxation on the surfaces, even for the so-called CT1 excitation. Since the two-coordinate potential surface in Fig. 3(d) is rather crude due to the rough assumptions for the relaxational modes, we next try full optimizations on the excited states. In Fig. 4(a), we show the behavior of the optimization starting from the CT2 excitation (the 23rd excited state) at ðd; sÞ ¼ ð0:07; 0:2Þ. We notice a further relaxation via a few level crossings, and the amount of relaxation is
more than 0.2 eV, measuring from the original point, i.e., ðd; sÞ ¼ ð0; 0Þ. In Fig. S3 of the Supplemental Material [18], we show the two structures before and after the optimization. Since it is a little difficult to see the structural difference there, we extract two quantities that characterize the structures. One is D, which averages three kinds of distances between the two inner molecules (see the Supplemental Material [18] for the details) and is normalized so as to give 0 and 1 for the LT and HT structures, respectively. On the other hand, s~ is defined as ðLT Þ= LT , with ¼ 1 þ 2 01 02 , and LT as that of the LT phase, where 1 and 2 are defined in Fig. 1(c) and the primed and unprimed variables stand for the bent and flat molecules, respectively. Plotting these quantities in Fig. 4(b), we clearly see that the optimization makes the structure even closer to that of HT. The whole view of this figure indicates that the relaxation for the CT2 state mainly occurs via the change of the D variable. In Fig. S4 of the Supplemental Material [18], we present the spectra for these configurations. They also show spectral change toward the metallic state or the (0101) CO state although the change is partial due to the structure fixed to the ground-state geometry in the outer two tetramers. From here on, we discuss the physical meanings of the results obtained so far and relate them with the actual material and its PIPT. First, we pay attention to the CT1 excitation. As was already mentioned partly with relation to Fig. 3, we do not find any substantial relaxation for this state in the present scheme that includes only the degrees of freedom within the tetramer. To confirm this situation, we also performed a full optimization for the CT1 excitation starting the optimizations from several possible structures such as ðd; sÞ ¼ ð0; 0Þ and (0.07, 0.2), but found no substantial relaxation. Furthermore, we have also tried another type of deformation pattern explicitly, i.e., that corresponding to (0101), although the structure goes back to the symmetric ones as far as we investigate in the present framework. It is therefore strongly suggested that other degrees of freedom such as those extended over several
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FIG. 5 (color online). Adiabatic potential curves deduced from the model Hamiltonian [Eq. (1) in Supplemental Material [18]]. For the solid lines, the site potentials i are kept at uniform values, while they are allowed to relax for the dotted lines.
tetramers and the motion of the anions are required to realize a PIPT via the CT1 excitation, and we return to this point later. Apart from our results for the photorelaxation, we make further comments on the CT1 excitation. Since its nature is the charge transfer between the inner and outer molecules, it is undoubtedly accompanied by valence changes in each molecule, as confirmed by the visualizations of the wave functions in Ref. [9]. This is also consistent with the large electron–‘molecular vibration’ coupling found particularly for the (þ 1, ) mode mentioned in Fig. 2(a). A question then arises: why do such valency changes not induce relaxation via a Holstein-type electron-lattice coupling? Our answer to this question is simply the smallness of its absolute value, in spite of the large frequency splitting. Namely, the magnitude of the coupling (the so-called g value) was estimated to be 0.08 eV from the frequency splitting [12]. This value is small and it only gives a relaxation energy less than 0.01 eV even in the localized limit, being further suppressed by a relatively large transfer energy of the order of 0.4 eV [10]. We also discuss the results described above from a more general point of view using a model Hamiltonian. The model that we adopt is a kind of the extended Peierls-Hubbard model and is expressed explicitly in the Supplemental Material [18]. It describes holes in the single chain of EDO-TTF molecules and takes account of its most relevant molecular orbital. The Hamiltonian includes an electronlattice coupling which varies the transfer energy between the neighboring molecules depending on the bond length. For the site energy l , we assume a pattern as (þ i , i , i , þi ) with i > 0 in the ith tetramer, reflecting the Coulombic potentials from the surrounding molecules including the anions [9]. The elastic energies of such degrees of freedom compose Henv . The actual calculation procedure
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is also explained in detail in the Supplemental Material [18]. Since our DFT treatment only considers the degrees of freedom in the tetramer, we first fix the value of i . The obtained potential curves are shown by the solid lines in Fig. 5 as a function of the size of a ‘‘domain.’’ Here the domain means an area with vanishing ql as defined by Eq. (2) in the Supplemental Material [18]. As seen in Fig. 5, a domain grows to an appreciable size only for the CT2 excitation. This feature is consistent with the result based on the DFT calculations. The dotted lines in the figure show the potential curves when the i values are allowed to vary in the course of relaxation. In this case, the channel of relaxation opens completely for the CT2 excitation and partly for the CT1 excitation. We note that the above change in the potential curves particularly for the CT2 excitation is based on the assumption that only the neighboring anions around the tetramer work as the source of the potential bias. In reality, the surrounding tetramers especially in different chains also contribute and are expected to need a longer time to make sufficient potential changes to realize the metallic state. Such a possible time delay gives us a picture that the PIPT in this material proceeds not monotonically but via a transient state such as that found by the present DFT calculation. In this respect, this transient state is also related to the newly found (0101)-type CO or charge fluctuation [17], since the former state resembles the latter state that will have equidistant EDO-TTF molecules and will necessarily appear after the melting of the tetramer structure. Lastly, we stress that the theoretical scheme described in this Letter is not specific to the photorelaxation in the EDO-TTF system. It can be applied to various PIPT phenomena and photorelaxation processes in the other materials such as molecular solids, metal compounds, and metal oxides. Excited-state molecular dynamics studies developed based on this work will provide a new theoretical tool for making molecular movies. This work is partly supported by Grant-In-Aid (No. 21540334) from JSPS.
[1] S. Koshihara and S. Adachi, J. Phys. Soc. Jpn. 75, 011005 (2006). [2] M. Lorenc, J. He´bert, N. Moisan, E. Trzop, M. Servol, M. Buron-Le Cointe, H. Cailleau, M. L. Boillot, E. Pontecorvo, M. Wulff, S. Koshihara, and E. Collet, Phys. Rev. Lett. 103, 028301 (2009). [3] H. Ichikawa, S. Nozawa, T. Sato, A. Tomita, K. Ichiyanagi, M. Chollet, L. Guerin1, N. Dean, A. Cavalleri, S. Adachi, T. Arima, H. Sawa, Y. Ogimoto, M. Nakamura, R. Tamaki, K. Miyano, and S. Koshihara, Nat. Mater. 10, 101 (2011). [4] G. Sciaini, M. Harb, S. G. Kruglik, T. Payer, C. T. Hebeisen, F.-J. M. Heringdorf, M. Yamaguchi, M. H. Hoegen, R. Ernstorfer, and R. J. Dwayne Miller, Nature (London) 458, 56 (2009).
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[5] M. Eichberger, H. Scha¨fer, M. Krumova, M. Beyer, J. Demsar, H. Beger, G. Moriena, G. Sciaini, and R. J. Dwayne Miller, Nature (London) 468, 799 (2010). [6] Y. Giret, A. Gelle´, and B. Arnaud, Phys. Rev. Lett. 106, 155503 (2011). [7] E. Papalazarou, J. Faure, J. Mauchain, M. Marsi, A. TalebIbrahimi, I. Reshetnyak, A. van Roekeghem, I. Timrov, N. Vast, B. Arnaud, and L. Perfetti, Phys. Rev. Lett. 108, 256808 (2012). [8] E. S. Zijlstra, L. L. Tatarinova, and M. E. Garcia, Phys. Rev. B 74, 220301(R) (2006). [9] K. Iwano and Y. Shimoi, Phys. Rev. B 77, 075120 (2008). [10] K. Iwano and Y. Shimoi, J. Phys. Soc. Jpn. 79, 103705 (2010). [11] A. Ota, H. Yamochi, and G. Saito, J. Mater. Chem. 12, 2600 (2002). [12] O. Drozdova, K. Yakushi, K. Yamamoto, A. Ota, H. Yamochi, G. Saito, H. Tashiro, and D. B. Tanner, Phys. Rev. B 70, 075107 (2004). [13] M. Chollet, L. Guerin, N. Uchida, S. Fukaya, H. Shimoda, T. Ishikawa, H. Yamochi, G. Saito, R. Tazaki, S. Adachi, and S. Koshihara, Science 307, 86 (2005).
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[14] N. Fukazawa, M. Shimizu, T. Ishikawa, Y. Okimoto, S. Koshihara, T. Hiramatzu, Y. Nakano, H. Yamochi, G. Saito, and K. Onda, J. Phys. Chem. C 116, 5892 (2012). [15] S. Ogihara, K. Onda, M. Shimizu, T. Ishikawa, Y. Okimoto, X. F. Shao, Y. Nakano, H. Yamochi, G. Saito, and K. Koshihara, J. Phys. Conf. Ser. 148, 012008 (2009). [16] O. Drozdova, K. Yakushi, A. Ota, H. Yamochi, and G. Saito, Synth. Met. 133–134, 277 (2003). [17] K. Onda, S. Ogihara, K. Yonemitsu, N. Maeshima, T. Ishikawa, Y. Okimoto, X. Shao, Y. Nakano, H. Yamochi, G. Saito, and S. Y. Koshihara, Phys. Rev. Lett. 101, 067403 (2008). [18] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.116401 for details of the DFT calculation scheme and the complete description of the model calculations. [19] K. Suzuki, K. Yamamoto, and K. Yakushi, Phys. Rev. B 69, 085114 (2004). [20] R. E. Stratmann, G. E. Scuseria, and M. J. Frisch, J. Chem. Phys. 109, 8218 (1998). [21] L. Lue¨r, C. Manzoni, G. Cerullo, G. Lanzani, and M. Meneghetti, Phys. Rev. Lett. 99, 027401 (2007).
116401-5
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PHYSICAL REVIEW LETTERS
Revealing the Photorelaxation Mechanism in a Molecular Solid Using Density-Functional Theory K. Iwano1 and Y. Shimoi2 1
Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), Graduate University for Advanced Studies, 1-1 Oho, Tsukuba 305-0801, Japan 2 Nanosystem Research Institute (NRI), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba 305-8568, Japan (Received 1 October 2012; revised manuscript received 12 January 2013; published 11 March 2013) Photorelaxation in a molecular crystal is investigated by a density-functional theory for the first time. A quasi-one-dimensional molecular compound, ðEDO-TTFÞ2 PF6 , is known to exhibit a photoinduced phase transition, which is characterized as a transition from a (0110)-type charge-ordering insulator phase to a high-temperature metallic phase. First, we apply the method of embedding a cluster in a self-consistent environment and succeed in constructing a stable tetramer structure of EDO-TTF molecules. The reliance of this cluster is ensured by a vibrational analysis that well reproduces the IR and Raman frequencies particularly for C¼C stretching modes including a relatively large degree of electron–‘molecular vibration’ coupling. Second, relaxations in the photoexcited states of this cluster are investigated by adiabatic potential-surface analyses and full structural optimization. A reaction coordinate is found to be quite unique for a relatively high-energy excitation, namely, the so-called CT2 excitation, which is interpreted as leading to the photoinduced phase transition. DOI: 10.1103/PhysRevLett.110.116401
PACS numbers: 71.15.Mb, 71.30.+h, 78.20.Bh
The physical and chemical aspects of photoinduced phase transitions (PIPTs) have been studied extensively from both the theoretical and experimental sides. In particular, one of the interests that have been continuing since the beginning of this subject is the visualization of their dynamical processes at the atomic level, namely, the making of ‘‘molecular movies’’ [1]. This viewpoint is regarded as particularly meaningful for compounds with complicated structures, for example, molecular solids, which have both intra- and intermolecular structures and are expected to exhibit a time evolution of the correlated degrees of freedom. Such a visualization is highly expected to reveal the mechanism of the PIPTs and give us new insight into the hidden nature of the materials as well as nonequilibrium processes. We already see several successful examples realized particularly by time-resolved x-ray or electron diffraction techniques in a variety of materials [2–5]. Motivated by such rapid developments in the experiments, we aim at theoretically visualizing photorelaxations in a molecular solid most realistically, based on a densityfunctional theory (DFT), and have succeeded in identifying a special path of structural relaxation. So far, most of the studies on this topic have been based on the so-called model calculations. There are a few studies using DFTs, although they are limited to a simple system like bismuth [6–8]. The advantage of the present method is its realistic treatment of electron-nuclei systems having pure Coulombic interactions, many orbitals, and many nuclei. In the PIPTs, or photorelaxation in a more general context, the essence of the dynamics lies in optically excited states and the structural deformations associated with them. 0031-9007=13=110(11)=116401(5)
For this reason, we apply a method of a cluster to this problem, instead of that based on a periodic boundary condition, which needs the treatment of a supercell and hence a relatively high computer load. Our cluster is augmented by a self-consistent environment that mimics the atmosphere around the cluster [9,10], as will be soon explained in detail, and the computer load is then similar to those in usual cluster calculations. Our special target, ðEDO-TTFÞ2 PF6 (EDO-TTF= ethylendioxy-tetrathiafulvalene), is a quasi-one-dimensional molecular solid having two thermal equilibrium phases: a high-temperature (HT) metallic phase and a lowtemperature (LT) insulator phase [11]. In the latter phase, the EDO-TTF molecules are characterized by two types of peculiar structural features, namely, tetramerization along the one-dimensional axis and intramolecular deformations. They induce a so-called (0110)-type charge ordering (CO), with 0 and 1 being the hole occupancies of the EDO-TTF molecules in the tetramer [11,12]. The PIPT observed in ðEDO-TTFÞ2 PF6 is basically a metallization starting from this state after a considerable time of at least 100 ps after photoexcitation [13,14]. This metallization is known to be induced by exciting both the lower and higher absorption bands at 0.5 and 1.5 eV [15], known respectively as the CT1 and CT2 bands [12,16]. Our finding here is that an intermolecular mode which increases the distance between the positively charged molecules is most deeply relevant in the metallization. Furthermore, this finding is also related to a newly found (0101)-type CO or charge fluctuation claimed to appear around 0.1 ps after the photoexcitaiton [17].
116401-1
Ó 2013 American Physical Society
To investigate the photorelaxations, we have to prepare a stable molecular cluster in its electronic ground state and exclude any instability that might exist already in the ground state. We hence define our self-consistent environment as the collection of point charges [9,10] and Lennard-Jones (LJ) 6–12 type of potential sources. More specifically, as is very schematically shown in Fig. 1(d), the cluster is surrounded by two layers; the closer layer consists of both the point charges and the LJ sources, while the farther one of only the point charges. The naming of ‘‘selfconsistent’’ means that the values of the point charges are determined self-consistently so as to match those calculated quantum mechanically within the system cluster. In Figs. 1(a) and 1(b), we compare the structures optimized by our cluster approach and determined by x-ray diffraction, respectively. In Fig. 1(a), within a cluster of three tetramers, the central tetramer is allowed to move freely. Note that the anions are kept at their observed positions and structures. Regarding the details of the calculation, see the Supplementary Material [18]. As is seen in the result, we consider that the present level of the optimized structure is practically acceptable. Looking more closely, we find that some changes occur as a result of the optimization. For example, the bending angles defined in Fig. 1(c) are changed by a few degrees for the inner two molecules while they are maintained in the outer two molecules. We also emphasize that the degree of the CO is also maintained: the molecular valencies in the tetramer, (0.18, 0.82, 0.82, 0.18), for the optimized structure are very similar to those for the observed structure, (0.17, 0.83, 0.83, 0.17).
Calculated vibrational frequencies provide further evidence that ensures the stability of the present cluster model. In Fig. 2. the theoretical results as well as the experimental ones [12] in the region of C¼C stretching modes are summarized with the conventional namings of the modes in Fig. 2(c) and an explicit tetramer structure in Fig. 2(d). We consider that the overall coincidence is satisfactory at this stage of the investigation. We note that the (0, ) mode is unexpectedly calculated at a frequency higher than 1600 cm1 , because of an inaccurate mixing with a hydrogen motion. Focusing on the physical aspect itself, the lowest mode, (þ 1, ), is worth special mention. As has been argued in the preceding studies for this [12] and other materials [19], the large frequency splitting between the Ag and Au modes originates from a rather strong electron–‘molecular vibration’ coupling via electron intermolecular transfer. This is the first reproduction of this coupling based on the DFT method, and again justifies the present building of the cluster. We now proceed to the investigation of the photorelaxations. Here, we relax only the central tetramer in the three consecutive tetramers, leaving the other two tetramers in the experimentally determined configuration, as is done in the ground-state optimization described in the previous paragraphs. Excited states are described by means of the time-dependent DFT method [20]. First, we analyze adiabatic potential surfaces with respect to the following two degrees of freedom. One is the intermolecular displace˚ , and defined as in Fig. 3(a). The other ment d in units of A is the degree of intramolecular deformation s that basically
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0,γ 0,β +1,α
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Au
+1,γ
FIG. 1 (color online). (a) and (b) Structures of three consecutive EDO-TTF tetramers optimized by our cluster approach (a) and determined by x-ray diffraction (b). (c) Definition of two bending angles. (d) Schematic picture of the two-layer model.
0,α
0 +1 +1 0 1400
Point Charges + LJ
0,γ 0,β +1,α
1500 1600 1700 Frequency (cm-1)
FIG. 2 (color online). Calculated (a) and observed (b) frequencies of the C¼C stretching modes. The assignments mean ‘‘(valency, mode).’’ (c) Illustration of three modes, and (d) the structure and the molecular valencies in a tetramer.
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FIG. 4 (color online). (a) Evolution of the CT2 excitation (the 23rd excited state) during the full optimization. (b) Summary of the structural changes expected for the CT2 excitation. The abbreviations, g-opt and ex-opt, are for the ground-state and excited-state optimized structures, respectively. The dotted arrow in (b) corresponds to the possible reaction path drawn by the solid line in Fig. 3(d).
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PHYSICAL REVIEW LETTERS
PRL 110, 116401 (2013)
1.2
s
0 1 2 3 4 5 6 7 8 -2
d (10 A)
FIG. 3 (color online). Definitions of (a) the ‘‘d’’ mode and (b) the ‘‘s’’ mode. The stick with an open circle stands for an EDO-TTF molecule. (c) Adiabatic potential curves for the lowest thirty excited states as a function of d. (d) Potential surface for the CT2 excitation with respect to d and s.
exchanges the bent and flat molecules between s ¼ 0 and s ¼ 2, as shown in Fig. 3(b). Note that the molecular shapes in the LT and HT phases are almost reproduced at s ¼ 0 and s ¼ 1, respectively. This deformation is created using the vibrational mode of an isolated molecule [21]. In Fig. 3(c), we draw the potential curves along the d axis, with s being fixed at zero. As is seen clearly, we find substantial relaxations for the state labeled CT2. This state has the largest oscillator strength in this energy region, and its largest component is the transition localized in the central tetramer, being consistent with the present structural change confined in the same tetramer. The meaning of this relaxation is well understood from the nature of the CT2 excitation, namely, a bonding to antibonding transition [9]. Moreover, its almost isolated nature is due to the exciton character from the lowest to the highest band typically seen in the three-quarter-filled band systems. Next, we draw the potential surface of this state using both of the variables in Fig. 3(d). The minimum in the surface is located around ðd; sÞ ¼ ð0:07; 0:2Þ, suggesting that the relaxation along the intramolecular deformation is not very effective. Regarding the other excitations, we do not recognize any substantial relaxation on the surfaces, even for the so-called CT1 excitation. Since the two-coordinate potential surface in Fig. 3(d) is rather crude due to the rough assumptions for the relaxational modes, we next try full optimizations on the excited states. In Fig. 4(a), we show the behavior of the optimization starting from the CT2 excitation (the 23rd excited state) at ðd; sÞ ¼ ð0:07; 0:2Þ. We notice a further relaxation via a few level crossings, and the amount of relaxation is
more than 0.2 eV, measuring from the original point, i.e., ðd; sÞ ¼ ð0; 0Þ. In Fig. S3 of the Supplemental Material [18], we show the two structures before and after the optimization. Since it is a little difficult to see the structural difference there, we extract two quantities that characterize the structures. One is D, which averages three kinds of distances between the two inner molecules (see the Supplemental Material [18] for the details) and is normalized so as to give 0 and 1 for the LT and HT structures, respectively. On the other hand, s~ is defined as ðLT Þ= LT , with ¼ 1 þ 2 01 02 , and LT as that of the LT phase, where 1 and 2 are defined in Fig. 1(c) and the primed and unprimed variables stand for the bent and flat molecules, respectively. Plotting these quantities in Fig. 4(b), we clearly see that the optimization makes the structure even closer to that of HT. The whole view of this figure indicates that the relaxation for the CT2 state mainly occurs via the change of the D variable. In Fig. S4 of the Supplemental Material [18], we present the spectra for these configurations. They also show spectral change toward the metallic state or the (0101) CO state although the change is partial due to the structure fixed to the ground-state geometry in the outer two tetramers. From here on, we discuss the physical meanings of the results obtained so far and relate them with the actual material and its PIPT. First, we pay attention to the CT1 excitation. As was already mentioned partly with relation to Fig. 3, we do not find any substantial relaxation for this state in the present scheme that includes only the degrees of freedom within the tetramer. To confirm this situation, we also performed a full optimization for the CT1 excitation starting the optimizations from several possible structures such as ðd; sÞ ¼ ð0; 0Þ and (0.07, 0.2), but found no substantial relaxation. Furthermore, we have also tried another type of deformation pattern explicitly, i.e., that corresponding to (0101), although the structure goes back to the symmetric ones as far as we investigate in the present framework. It is therefore strongly suggested that other degrees of freedom such as those extended over several
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Excitation Energy (eV)
2.5
CT2
2.0
1.5
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CT1
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10
20
30
40
Domain Size (sites)
FIG. 5 (color online). Adiabatic potential curves deduced from the model Hamiltonian [Eq. (1) in Supplemental Material [18]]. For the solid lines, the site potentials i are kept at uniform values, while they are allowed to relax for the dotted lines.
tetramers and the motion of the anions are required to realize a PIPT via the CT1 excitation, and we return to this point later. Apart from our results for the photorelaxation, we make further comments on the CT1 excitation. Since its nature is the charge transfer between the inner and outer molecules, it is undoubtedly accompanied by valence changes in each molecule, as confirmed by the visualizations of the wave functions in Ref. [9]. This is also consistent with the large electron–‘molecular vibration’ coupling found particularly for the (þ 1, ) mode mentioned in Fig. 2(a). A question then arises: why do such valency changes not induce relaxation via a Holstein-type electron-lattice coupling? Our answer to this question is simply the smallness of its absolute value, in spite of the large frequency splitting. Namely, the magnitude of the coupling (the so-called g value) was estimated to be 0.08 eV from the frequency splitting [12]. This value is small and it only gives a relaxation energy less than 0.01 eV even in the localized limit, being further suppressed by a relatively large transfer energy of the order of 0.4 eV [10]. We also discuss the results described above from a more general point of view using a model Hamiltonian. The model that we adopt is a kind of the extended Peierls-Hubbard model and is expressed explicitly in the Supplemental Material [18]. It describes holes in the single chain of EDO-TTF molecules and takes account of its most relevant molecular orbital. The Hamiltonian includes an electronlattice coupling which varies the transfer energy between the neighboring molecules depending on the bond length. For the site energy l , we assume a pattern as (þ i , i , i , þi ) with i > 0 in the ith tetramer, reflecting the Coulombic potentials from the surrounding molecules including the anions [9]. The elastic energies of such degrees of freedom compose Henv . The actual calculation procedure
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is also explained in detail in the Supplemental Material [18]. Since our DFT treatment only considers the degrees of freedom in the tetramer, we first fix the value of i . The obtained potential curves are shown by the solid lines in Fig. 5 as a function of the size of a ‘‘domain.’’ Here the domain means an area with vanishing ql as defined by Eq. (2) in the Supplemental Material [18]. As seen in Fig. 5, a domain grows to an appreciable size only for the CT2 excitation. This feature is consistent with the result based on the DFT calculations. The dotted lines in the figure show the potential curves when the i values are allowed to vary in the course of relaxation. In this case, the channel of relaxation opens completely for the CT2 excitation and partly for the CT1 excitation. We note that the above change in the potential curves particularly for the CT2 excitation is based on the assumption that only the neighboring anions around the tetramer work as the source of the potential bias. In reality, the surrounding tetramers especially in different chains also contribute and are expected to need a longer time to make sufficient potential changes to realize the metallic state. Such a possible time delay gives us a picture that the PIPT in this material proceeds not monotonically but via a transient state such as that found by the present DFT calculation. In this respect, this transient state is also related to the newly found (0101)-type CO or charge fluctuation [17], since the former state resembles the latter state that will have equidistant EDO-TTF molecules and will necessarily appear after the melting of the tetramer structure. Lastly, we stress that the theoretical scheme described in this Letter is not specific to the photorelaxation in the EDO-TTF system. It can be applied to various PIPT phenomena and photorelaxation processes in the other materials such as molecular solids, metal compounds, and metal oxides. Excited-state molecular dynamics studies developed based on this work will provide a new theoretical tool for making molecular movies. This work is partly supported by Grant-In-Aid (No. 21540334) from JSPS.
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