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Fractional dimension analysis of self-trapped Frenkel excitons in molecular microcrystallites
Journal of Luminescence 87}89 (2000) 263}265
Fractional dimension analysis of self-trapped Frenkel excitons in molecular microcrystallites Masumi Takeshima *, Ken-ichi Mizuno, Atsuo H. Matsui Organo-Optic Research Laboratory, 48-21 Bandoujima, Kitagou, Katsuyama, Fukui 911-0056, Japan Department of Physics, Konan University, Okamoto, Kobe 658-0072, Japan
Abstract The concept of fractional dimension is introduced to analyze self-trapped exciton characteristics in organic crystallites of various shapes since the concept of the integral dimension is not always applied satisfactorily to crystallites of "nite side lengths. The theory is based on the tight-binding approach. It is shown that anomaly in self-trapping appears when the fractional dimension is close but a little larger than the integral dimension. 2000 Elsevier Science B.V. All rights reserved. Keywords: Self-trapped excitons; Crystallite shape e!ect; Fractional dimension
1. Introduction Self-trapped exciton (STE) problem is one of the attractive current themes because the exciton}phonon interaction is a key factor a!ecting stability of STE. It was recently reported [1,2] that stability of STE also depends strongly on crystallite shape. Hitherto integral dimensions, one dimension (1D), two dimension (2D) and three dimension (3D), have been considered on crystallites. However, clearly the concept of the integral dimension is not always realistic, because in fabrication of crystallites, they grow in various shapes with "nite sizes [3]. To understand stability of STE in crystallites it is therefore useful to study the STE character in connection with crystallite shape for which the concept of the intermediate dimension is appropriate. In this paper we introduce the concept of the fractional dimension for representing the shape of crystallites and then demonstrate how one can "nd the magnitude of the fractional dimension of speci"c crystallites of parallelepiped shapes with various side lengths. It is found that anomaly in exciton self-trapping appears when the mag-
* Corresponding author. Tel./fax: #81-779-89-1336. E-mail address: [email protected] (M. Takeshima)
nitude of the fractional dimension is close but a little larger than the integral dimension.
2. Theory Let us consider a parallelepiped whose lattice numbers are N , N and N along the x, y and z V W X directions, respectively. We assume the nearest-neighbor interaction only in each direction, and assume that physical properties along the crystallographic directions are the same for all three directions. Then applying a method used in Ref. [2], we obtain the STE energy as E (b , b , b ), where b (k"x, y, z) is a variational V W X I parameter, with which the minimum of E is found at b "b . E is given in terms of two material I I parameters c"(= /=)B/(2Mu ) and h"! D =/(= B). Here B""2"=" is the half-width of the exciton band of the bulk crystal, = the transfer matrix between the nearest-neighbor lattices, D the site shift energy, = "d=/da and D "dD/da, where a is the lattice constant in each direction and Mu the force constant with the mass of the lattice molecule M and the vibrating frequency u /(2p). We assume the microcrys tallites to be a direct transition material, i.e., =(0. Note that E (0, 0, 0) should be the lowest energy of the free exciton (FE) band.
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 0 9 - 9
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M. Takeshima et al. / Journal of Luminescence 87}89 (2000) 263}265
The self-trap depth, E , is de"ned as E "E (0, 0, 0)!E (b , b , b ). (1) V W X By considering a cube (3D), a square (2D) and a line (1D) with equal side lengths, i.e., N "N and equal I physical properties in any directions, the energy of the self-trapped exciton state in the d dimension (d"3, 2, 1) systems has been given in Ref. [2] as E (b)"Bd+K (b)#K (b)[X(b)]B\,. (2) Here K (b), K (b) and X(b), which is independent of d, are de"ned in Ref. [2]. Now, we extend this dimension, d, to the fractional dimension, assuming that Eq. (2) is applicable also to non-integer d. b is a parameter with which the minimum of Eq. (2) is found at b"b . The self-trap depth is
de"ned as E "E (0)!E (b ). (3) The fractional dimension, d, can be found, using Eq. (3) as the basis.
3. Results and discussion Fig. 1 shows the E versus c relation with N as a X parameter. For a given N and a given h there is a threX shold value of c, c , below which the STE state does not exist. The positions of c are shown by arrows. Similarly, for a given N and a given c we have a threshold value of X h, h , below which the STE state does not exist [4]. We determine fractional dimension, d, through c . We seek the d value for which the c value found from Eq. (3) with N"R agrees with the c found from Eq. (1) for an arbitrary parallelepiped crystallite with lattice numbers N , N and N . V W X Fig. 2 shows the result thus obtained. It shows the fractional dimension, d, plotted against a normalized lattice number N . The de"nition of N is as follows: VWX VWX
Fig. 1. E as a function of c with N as a parameter for X crystallites with h"1 and N "N "R, representing V W the 2D}3D system. Arrows indicate the threshold positions of c, i.e., c .
Fig. 2. The fractional dimension, d, determined using Eq. (3) for results of c calculated from Eq. (1). We de"ne a structure N "N /100 and N "N "R in the 2D}3D zone (䉱), X VWX V W a structure N "N , N "R and N "1 in the 1D}2D zone W VWX V X (䊏), and a structure N "100N and N "N "1 in the V VWX W X 0D}1D zone (䉬).
(1) N "N /100 with "xed values N "N "R for X VWX V W crystallites between 2D and 3D, (2) N "N with "xed W VWX values N "R and N "1 for crystallites between 1D V X and 2D, and (3) N "100N with "xed values V VWX N "N "1 for crystallites between 0D and 1D. d is W X equal to 3, 2, and 1, respectively, at the right and left edges of the 2D}3D zone, and at the left edge of the 1D}2D zone. The curve is monotonic except at N " VWX 200 (N "2, N "N "R) and at N "0.04 (N "4, X V W VWX V N "N "1), where we "nd exceptionally large d's W X as compared to those for neighbors. The fact that d" 0 at N "0.03 indicates that the STE state does not VWX exist at N "0.02 (N "2, N "N "1) and at VWX V W X N "0.01 (N "N "N "1). The last result is natural VWX V W X since the point N "0.01 means an isolated molecule VWX (or atom). The anomaly at N "200 (N "2, N "N "R) is VWX X V W re#ected in the E }c relation, which is shown in Fig. 1 for the 2D}3D zone. E decreases monotonically with in creasing N in the range N 53, though only several X X curves are shown for simplicity. In contrast, for N from X 1 to 3, E increases with increasing N . X Similar discussion is applied to the 1D}2D zone in the E }c relation under N "R and N "1 with N as V X W a varying parameter. We found an anomalous behavior in the E }c relation around N "2 (N "R, N "1). W V X However, no anomaly is found in the 1D}2D zone in the d}N relation (ref. Fig. 2). VWX For the 0D}1D zone, anomaly is found in the d}N VWX relation in Fig. 2 around N "0.04 (N "4, N " VWX V W N "1) , but no anomaly was found in the E }c relation X as is found in Fig. 1. The E }c relation in the 0D}1D zone is shown in Fig. 3. In Fig. 3 E decreases with decreasing N , indicating that the STE is destabilized with decreasing V crystallite size in the 0D}1D zone. We do not have curves for N 42 because we have no STE state for N 42. The V X
M. Takeshima et al. / Journal of Luminescence 87}89 (2000) 263}265
265
self-trapping are found for crystallites with fractional dimensions a little larger than the nearest integral dimensions. We have discussed STE stability along the crystallite size reduction in the sequential order 3DP2DP1DP 0D. However, we can consider other paths of shape changes, such as direct change from 3D to 0D or a direct change from 3D to 1D. Discussion associated with those direct shape changes will be given elsewhere [4]. Fig. 3. E as a function of c with N as a parameter for V crystallites with h"1 and N "N "1, representing the W X 0D}1D system.
stability of STE in the 0D}1D zone are, therefore, quite di!erent from those in the 1D}2D and 2D}3D zones. Summarizing our results, anomalies appear in the d}N relation when N "200 (N "2, N "N " VWX VWX X V W R) and N "0.04 (N "4, N "N "1) . In the VWX V W X E }c relation, anomaly is found at N "2 (N " VWX W 2, N "R, N "1). These three anomalies correspond V X to the exciton localization in (1) a thin crystallite made of two layers (N "200; N "2, N "N "R), (2) a VWX X V W crystallite made of two molecular chains (N "2, N "1, W V N "R), and (3) a linear molecular chain made of X four molecules (N "0.04; N "4, N "N "1). It VWX V W X should be emphasized that these anomalies in exciton
Acknowledgements This work is partly supported by a Grant-in-Aid for Scienti"c Research on Priority Area (B) on `Laser Chemistry of Single Nanometer Organic Particlesa from the Ministry of Education, Sports and Culture of Japanese Government (10207206).
References [1] E.I. Rashba, Synthetic Met. 64 (1994) 255. [2] M. Takeshima, J. Singh, A.H. Matsui, Chem. Phys. 233 (1998) 97. [3] A.H. Matsui, K. Mizuno, O. Nishi, Y. Matsushima, M. Shimizu, T. Goto, M. Takeshima, Chem. Phys. 194 (1995) 167. [4] M. Takeshima, A.H. Matsui, unpublished.
Fractional dimension analysis of self-trapped Frenkel excitons in molecular microcrystallites Masumi Takeshima *, Ken-ichi Mizuno, Atsuo H. Matsui Organo-Optic Research Laboratory, 48-21 Bandoujima, Kitagou, Katsuyama, Fukui 911-0056, Japan Department of Physics, Konan University, Okamoto, Kobe 658-0072, Japan
Abstract The concept of fractional dimension is introduced to analyze self-trapped exciton characteristics in organic crystallites of various shapes since the concept of the integral dimension is not always applied satisfactorily to crystallites of "nite side lengths. The theory is based on the tight-binding approach. It is shown that anomaly in self-trapping appears when the fractional dimension is close but a little larger than the integral dimension. 2000 Elsevier Science B.V. All rights reserved. Keywords: Self-trapped excitons; Crystallite shape e!ect; Fractional dimension
1. Introduction Self-trapped exciton (STE) problem is one of the attractive current themes because the exciton}phonon interaction is a key factor a!ecting stability of STE. It was recently reported [1,2] that stability of STE also depends strongly on crystallite shape. Hitherto integral dimensions, one dimension (1D), two dimension (2D) and three dimension (3D), have been considered on crystallites. However, clearly the concept of the integral dimension is not always realistic, because in fabrication of crystallites, they grow in various shapes with "nite sizes [3]. To understand stability of STE in crystallites it is therefore useful to study the STE character in connection with crystallite shape for which the concept of the intermediate dimension is appropriate. In this paper we introduce the concept of the fractional dimension for representing the shape of crystallites and then demonstrate how one can "nd the magnitude of the fractional dimension of speci"c crystallites of parallelepiped shapes with various side lengths. It is found that anomaly in exciton self-trapping appears when the mag-
* Corresponding author. Tel./fax: #81-779-89-1336. E-mail address: [email protected] (M. Takeshima)
nitude of the fractional dimension is close but a little larger than the integral dimension.
2. Theory Let us consider a parallelepiped whose lattice numbers are N , N and N along the x, y and z V W X directions, respectively. We assume the nearest-neighbor interaction only in each direction, and assume that physical properties along the crystallographic directions are the same for all three directions. Then applying a method used in Ref. [2], we obtain the STE energy as E (b , b , b ), where b (k"x, y, z) is a variational V W X I parameter, with which the minimum of E is found at b "b . E is given in terms of two material I I parameters c"(= /=)B/(2Mu ) and h"! D =/(= B). Here B""2"=" is the half-width of the exciton band of the bulk crystal, = the transfer matrix between the nearest-neighbor lattices, D the site shift energy, = "d=/da and D "dD/da, where a is the lattice constant in each direction and Mu the force constant with the mass of the lattice molecule M and the vibrating frequency u /(2p). We assume the microcrys tallites to be a direct transition material, i.e., =(0. Note that E (0, 0, 0) should be the lowest energy of the free exciton (FE) band.
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 0 9 - 9
264
M. Takeshima et al. / Journal of Luminescence 87}89 (2000) 263}265
The self-trap depth, E , is de"ned as E "E (0, 0, 0)!E (b , b , b ). (1) V W X By considering a cube (3D), a square (2D) and a line (1D) with equal side lengths, i.e., N "N and equal I physical properties in any directions, the energy of the self-trapped exciton state in the d dimension (d"3, 2, 1) systems has been given in Ref. [2] as E (b)"Bd+K (b)#K (b)[X(b)]B\,. (2) Here K (b), K (b) and X(b), which is independent of d, are de"ned in Ref. [2]. Now, we extend this dimension, d, to the fractional dimension, assuming that Eq. (2) is applicable also to non-integer d. b is a parameter with which the minimum of Eq. (2) is found at b"b . The self-trap depth is
de"ned as E "E (0)!E (b ). (3) The fractional dimension, d, can be found, using Eq. (3) as the basis.
3. Results and discussion Fig. 1 shows the E versus c relation with N as a X parameter. For a given N and a given h there is a threX shold value of c, c , below which the STE state does not exist. The positions of c are shown by arrows. Similarly, for a given N and a given c we have a threshold value of X h, h , below which the STE state does not exist [4]. We determine fractional dimension, d, through c . We seek the d value for which the c value found from Eq. (3) with N"R agrees with the c found from Eq. (1) for an arbitrary parallelepiped crystallite with lattice numbers N , N and N . V W X Fig. 2 shows the result thus obtained. It shows the fractional dimension, d, plotted against a normalized lattice number N . The de"nition of N is as follows: VWX VWX
Fig. 1. E as a function of c with N as a parameter for X crystallites with h"1 and N "N "R, representing V W the 2D}3D system. Arrows indicate the threshold positions of c, i.e., c .
Fig. 2. The fractional dimension, d, determined using Eq. (3) for results of c calculated from Eq. (1). We de"ne a structure N "N /100 and N "N "R in the 2D}3D zone (䉱), X VWX V W a structure N "N , N "R and N "1 in the 1D}2D zone W VWX V X (䊏), and a structure N "100N and N "N "1 in the V VWX W X 0D}1D zone (䉬).
(1) N "N /100 with "xed values N "N "R for X VWX V W crystallites between 2D and 3D, (2) N "N with "xed W VWX values N "R and N "1 for crystallites between 1D V X and 2D, and (3) N "100N with "xed values V VWX N "N "1 for crystallites between 0D and 1D. d is W X equal to 3, 2, and 1, respectively, at the right and left edges of the 2D}3D zone, and at the left edge of the 1D}2D zone. The curve is monotonic except at N " VWX 200 (N "2, N "N "R) and at N "0.04 (N "4, X V W VWX V N "N "1), where we "nd exceptionally large d's W X as compared to those for neighbors. The fact that d" 0 at N "0.03 indicates that the STE state does not VWX exist at N "0.02 (N "2, N "N "1) and at VWX V W X N "0.01 (N "N "N "1). The last result is natural VWX V W X since the point N "0.01 means an isolated molecule VWX (or atom). The anomaly at N "200 (N "2, N "N "R) is VWX X V W re#ected in the E }c relation, which is shown in Fig. 1 for the 2D}3D zone. E decreases monotonically with in creasing N in the range N 53, though only several X X curves are shown for simplicity. In contrast, for N from X 1 to 3, E increases with increasing N . X Similar discussion is applied to the 1D}2D zone in the E }c relation under N "R and N "1 with N as V X W a varying parameter. We found an anomalous behavior in the E }c relation around N "2 (N "R, N "1). W V X However, no anomaly is found in the 1D}2D zone in the d}N relation (ref. Fig. 2). VWX For the 0D}1D zone, anomaly is found in the d}N VWX relation in Fig. 2 around N "0.04 (N "4, N " VWX V W N "1) , but no anomaly was found in the E }c relation X as is found in Fig. 1. The E }c relation in the 0D}1D zone is shown in Fig. 3. In Fig. 3 E decreases with decreasing N , indicating that the STE is destabilized with decreasing V crystallite size in the 0D}1D zone. We do not have curves for N 42 because we have no STE state for N 42. The V X
M. Takeshima et al. / Journal of Luminescence 87}89 (2000) 263}265
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self-trapping are found for crystallites with fractional dimensions a little larger than the nearest integral dimensions. We have discussed STE stability along the crystallite size reduction in the sequential order 3DP2DP1DP 0D. However, we can consider other paths of shape changes, such as direct change from 3D to 0D or a direct change from 3D to 1D. Discussion associated with those direct shape changes will be given elsewhere [4]. Fig. 3. E as a function of c with N as a parameter for V crystallites with h"1 and N "N "1, representing the W X 0D}1D system.
stability of STE in the 0D}1D zone are, therefore, quite di!erent from those in the 1D}2D and 2D}3D zones. Summarizing our results, anomalies appear in the d}N relation when N "200 (N "2, N "N " VWX VWX X V W R) and N "0.04 (N "4, N "N "1) . In the VWX V W X E }c relation, anomaly is found at N "2 (N " VWX W 2, N "R, N "1). These three anomalies correspond V X to the exciton localization in (1) a thin crystallite made of two layers (N "200; N "2, N "N "R), (2) a VWX X V W crystallite made of two molecular chains (N "2, N "1, W V N "R), and (3) a linear molecular chain made of X four molecules (N "0.04; N "4, N "N "1). It VWX V W X should be emphasized that these anomalies in exciton
Acknowledgements This work is partly supported by a Grant-in-Aid for Scienti"c Research on Priority Area (B) on `Laser Chemistry of Single Nanometer Organic Particlesa from the Ministry of Education, Sports and Culture of Japanese Government (10207206).
References [1] E.I. Rashba, Synthetic Met. 64 (1994) 255. [2] M. Takeshima, J. Singh, A.H. Matsui, Chem. Phys. 233 (1998) 97. [3] A.H. Matsui, K. Mizuno, O. Nishi, Y. Matsushima, M. Shimizu, T. Goto, M. Takeshima, Chem. Phys. 194 (1995) 167. [4] M. Takeshima, A.H. Matsui, unpublished.