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Nonparallel transition dipole moments and the polarization dependence of electroabsorption in nonoriented conjugated polymer films
11 February 1994
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 2 18 ( 1994) 195 199
Nonparallel transition dipole moments and the polarization dependence of electroabsorption in nonoriented conjugated polymer films T.W. Hagler Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 20 October 1993; in tinal form 29 November 1993
Abstract We present a comprehensive analysis of the polarization dependence of the field-induced absorption in nonoriented conjugated polymer films. The results demonstrate that the polarization anisotropy of the field-induced absorption is directly related to the relative angle, A&, between the dominant one-photon and two-photon transition dipole moments responsible for the nonlinear effect. The polarization anisotropy has a maximum value of 3 : 1 for parallel transition dipole moments, and a minimum value of 1: 3 for perpendicular transition dipole moments. We demonstrate that the predicted field-induced absorption anisotropy is consistent with Kleinmann symmetry for any relative orientation of the dominant transition dipole moments.
1. Introduction Electric field modulation spectroscopy (EFMS) has recently become an important technique for investigating the electronic structure of nonlinear organic chromophores. One aspect of EFMS, which is often overlooked, is that the technique involves the mixing of two independently polarized electric fields, i.e. the optical field and the applied dc electric field. As a consequence, one can determine two independent components of the macroscopic nonlinear susceptibility tensor by measuring the field-induced absorption or reflectance for optical polarization parallel and perpendicular to the applied dc electric field. By analyzing the ratio of these two components, one can then extract information pertaining to the microscopic electronic structure of the material. In a recently submitted article, Hagler, Pakbaz and Heeger [ 1 ] demonstrated that the polarization dependence of the field-induced absorption in oriented
conjugated polymer samples is related to the angle of the dominant G to ‘9%”transition dipole moment with respect to the drawing axis, and that in certain materials, this angle is related to the spatial extent of the excited state wavefunction. In contrast, the polarization anisotropy of the nonoriented sample, defined as the ratio of the field-induced absorption for optical polarization parallel and perpendicular to the applied dc electric field, has not been sufficiently examined. Initial investigations suggested that the polarization dependence of the field-induced absorption in nonoriented conjugated polymer films was also related to the off-axis component of the transition dipole moment [ 2,3 1. However, this analysis was based on the 22 = 4 projections of the optical and dc electric field intensities onto the polymer chain axis, rather than the 42 = 16 projections of the electric field amplitudes. By considering the various projections of the optical and dc electric field amplitudes onto the poly-
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI0009-2614(93)El486-Z
196
T. W. Hagler / Chem. Phys. Letters 218 (1994) 195-199
mer chain axis and then averaging over all possible chain orientations, one obtains an expression for the polarization dependence of the field-induced absorption that is independent of the angle the transition dipole moment makes with respect to the polymer chain axis. In fact, any theoretical model (see for instance ref. [4] ), that assumes that the dominant transition dipole moments are parallel will predict a field-induced absorption polarization anisotropy ratio of 3 : 1, regardless of the angle the transition dipole moments make with respect to the polymer chain axis. In nonoriented conjugated polymer samples, the polarization dependence of the field-induced absorption routinely violates this predicted anisotropy. In this Letter, we examined the polarization dependence of the field-induced absorption for an isotropic distribution of quasi-one-dimensional nonlinear chromophores of Cl,, spatial symmetry. Our results show that the polarization dependence of the macroscopic nonlinear susceptibility, xf.&( w, 0,O) depends critically on the details of the microscopic nonlinear polarizability, y&a (w, 0,O). In particular, the observed polarization dependence of the tield-induced absorption is directly related to the relative angle, A&, between the two dominant transition dipole moments responsible for the nonlinear effect (‘9, Ir IG) and ( “4 ( r I ‘%I,), respectively. The polarization anisotropy has a maximum value of 3 : 1 for parallel transition dipole moments, A&=0, and a minimum value of I : 3 for perpendicular transition dipole moments, A&=x/2. In addition, we calculate several components of the macroscopic nonlinear susceptibility tensor x1,&( w, 0,O) to demonstrate that the field-induced absorption anisotropy is consistent with Kleinmann symmetry for any value of the relative angle A&.
involved in the nonlinear response. To simplify the discussion, we use the geometry defined in Fig. 1 and assume that the applied dc electric field I;is along the X axis, and that the incident optical field E( k, w) is propagating in the 2 direction and is polarized at an angle of 8 with respect to the electric field. The distribution of polymer chains is assumed to be random in three dimensions; however, the plane defined by the optical polarization and the electric field polarization reduces the problem to two dimensions. The nonlinear wave equation describes the propagation and generation of electromagnetic fields in the system due to the applied dc electric field perturbation and the incident optical field, and is, therefore, the most general description of electro-optic phenomena, describing both electroabsorption and electroreflectance. In this analysis we will focus exclusively on the field-induced absorption; the corresponding expressions for the electroreflectance can be obtained by applying the appropriate boundary conditions. Using the definitions of the complex linear dielectric function, Z= 1+4x~(‘)=N& and the complex linear refractive index, No = no+ iKo,the nonlinear wave equation describing the modulation of the optical field by the applied dc electric field can be written as
E(k.a)
X, F
2. Analysis of the nonlinear response 2.1. Polarization dependence of-AT/T(B)
In this section we examine the polarization dependence of the measured nonlinear response -AT/ T( 0) with respect to the applied electric field. The objective is to demonstrate that the observed polarization dependence is related to the relative angle A&, between the two dominant transition dipole moments
Fig. 1. The geometry used to analyze the polarization dependence of the field-induced absorption. The upper-case coordinates define the axis of the macroscopic linear and nonlinear susceptibility, and in amorphous, or polycrystalline materials, are dictated by sample geometry. The lower-case coordinates define the axis of the microscopic linear and nonlinear polarizability, and refer to the axis of particular polymer chain.
T. W. Hagier /Chem. Phys. titters 218 (1994) 195-199
x(F-.l)(F-R)(&L)eiZn']E(k,o,z)=O,
(1)
where Zc,= w/c is the free space wavevector of the optical field, and $2=0 is the frequency of the applied dc electric field. In writing Eq. ( 1)) we have used the experimental observation that the electroabsorption lineshape tends to resemble a positive derivative of the unperturbed absorption coefficient. This dictates that the subscripts Z and L refer to the matrix elements associated with the optical field and J and K refer to matrix elements associated with the applied dc electric field [ 5 1. Note that the upper-case coordinates Z, J, K and L refer to the macroscopic coordinate system of the sample, and that the macroscopic linear and nonlinear polarization P(I) and PC3) are obtained by a vector summation over p(l) and c3) the microscopic linear and nonlinear polarizaP 3 tions of the individual polymer chains. Hence, in nonoriented samples, the subscripts on xf& have nothing to do with the polymer chain axis, but are dictated by sample geometry. For the sample geometry shown in Fig. 1, it is convenient to choose Z, J, K, L=-lx r>. By integrating the solution of Eq. ( 1) over the thickness of the sample, we obtain the experimentally measured nonlinear response -~(W,e)-dAOl(W,8)eizn’,
(2)
where Acy (w, .9) is defined as the polarization-dependent field-induced absorption coefficient, which can be written in terms of the macroscopic nonlinear susceptibility tensor as Aa(w,
4x 0) =k, -Im[ZCWL(o, ,N0 ,2
+ZV:#&(o,
0, 0)sin2(@]F2.
O,O)cosW (3)
Because of the isotropic distribution of the conjugated polymer chains, the linear optical coefficients are polarization independent. Hence, the polarization dependence of the measured nonlinear response, - AT/ T( t9) is contained entirely in the third-order nonlinear susceptibilityX(3’ (0, 0,0), which, in turn, depends on the details of the microscopic nonlinear
197
polarizability. Therefore, to understand the physical origin of the polarization dependence of the tield-induced absorption in a macroscopic sample, we need to consider the microscopic polarizability of the individual polymer chains. The nonlinear response of an individual chain is proportional to the third-order polarizability tensor y& (0, 0, 0 ), where the indices i, j, k, I are defined in a local coordinate system with respect to the polymer chain axis. Since there are two independently polarized electric fields, E( k, w) and F, the 8 component of the hyperpolarizability of a polymer chain oriented along the C#I axis is given by the various projections of the optical field and the applied dc electric field onto the unit vectors i, j, k, 1. As stated previously, the observed electroabsorption lineshape dictates that the subscripts i and 1refer to the matrix elements associated with the optical field and j and k refer to matrix elements associated with the applied dc electric field [ 51. As a result, there are 42 = 16 independent terms in the microscopic nonlinear polarizability which contribute to the macroscopic nonlinear susceptibility. The final expression for the macroscopic nonlinear susceptibility of the nonoriented sample is obtained by averaging over all possible chain orientations,
x~~~(w,O,O)=tl~ij~~~~y~~~(w,O,O) ,,, , x(i-4)ci’f)(M)(m))
(4)
where the lower-case subscripts x and y refer to directions parallel and perpendicular to the molecular axis of the polymer chain, respectively. Note that possible rotations about the polymer chain axis are incorporated into the averaged perpendicular coordinate y. By inserting Eq. (4) into Eq. (3) we obtain the following polarization dependence for the lield-induced absorption of the nonoriented sample:
- A+,C+&A~,,,(w){[3-2cos2(A&)] -cos2(0)[2-4cos2(A&)]),
(5)
where A@*is the relative angle between the dominant G to ‘9, and ‘B, to “‘4 transition dipole moments,
T. W. Hagler / Chem. Phys. Letters 218 (1994) 195-199
198
(6) and
where rlf)(o, 0,O) is the component of the microscopic nonlinear polarizability associated with the optical polarization parallel to the G to kB, transition dipole moment, and the electric field parallel to the ‘%, to “‘A, transition dipole moment (i.e. E(k, w) and Fare along the $A and aji axis, respectively). It is clear from Eq. (5) that if the two dominant transition dipole moments responsible for the fieldinduced absorption are parallel, A& = 0, we expect to see a polarization anisotropy of 3 : 1 in the nonlinear response. If, on the other hand, the two dominant transition dipole moments are perpendicular, A&=x/2, we expect to see an anisotropy of 1: 3 in the nonlinear response. To facilitate the analysis of experimental data, Eq. (5) can be inverted to yield the relative angle between the dominant transition dipole moments as a function of the measured anisotropy 3-P COS~(A@~)= 2+2p’ where p= AT( 90” ) /AT( 0” ). Since deviations from the 3 : 1 ratio predicted for parallel transition dipole moments are common in the literature [ 2,3,6,7], the physical origin of the nonparallel transition dipole moments becomes an essential ingredient to a more complete description of the nonlinear response. 2.2. Kleinmann
symmetry
It has recently been suggested [ 81 that the existence of low-lying perpendicular electronic bands leads to a violation of Kleinmann symmetry [ 9, lo], which predicts the following relationship between the various components of the nonlinear susceptibility tensor:
In order to investigate possible violations of Kleinmann symmetry within our model for the polarization dependence of the field-induced absorption, which includes arbitrarily nonparallel transition dipole moments, we generate the following components of the third-order nonlinear susceptibility tensor:
xBn4w
090)
= &YIF’( w,O,O)[-3+4cos2(A@~)], Xlr33rx(~, O,O) 1 = ,,rl;‘(
XGMW
0,0,0)[3-2cos2(A&],
090) = &F’cw,
0, 0) *
(10)
From Eq. ( lo), we see that our model for the polarized field-induced absorption is consistent with Kleinmann symmetry for any value of the relative angle A&.
3. Conclusions We have reexamined the analysis of the polarization dependence of the field-induced absorption for nonoriented quasi-one-dimensional samples and found a discrepancy with previously published results, which considered only the projections of the optical and dc electric field intensities onto the polymer chain axis. Our analysis, which considers the projections of the optical and dc electric field amplitudes onto the polymer chain axis, relates the observed anisotropy of the field-induced absorption in the nonoriented material to the relative angle A@, between the dominant G to 9” and ‘%, to “AB transition dipole moments responsible for the nonlinear effect. The polarization anisotropy of the field-induced absorption in nonoriented samples has a maximum value of 3 : 1 for parallel transition dipole moments, Agd=O, and 1: 3 for perpendicular transition dipole moments, A& = ~12. The results are independent of the localization length of the excited state wavefunc-
T. W. Hagler / Chem. Phys. Letters 218 (1994) 195-199
tion, or the angle that the transition dipole moments make with respect to the molecular axis. It is clear that by understanding the origin of offaxis and nonparallel transition dipole moments, one would achieve a more comprehensive picture of the nonlinear hyperpolarizability tensor, r&i], in quasione-dimensional materials such as polyacetylene and poly(pphenylenevinylene). In the absence of disorder, the relative angle between the dominant G to ‘B, and “B, to “‘A, transition dipole moments is a fundamental quantity of the material which should not depend on the parameters of a particular model. Therefore a direct measurement of A& would be a valuable asset in determining which theoretical models are best at describing a particular system. Because of its simplicity, polarized EFMS is an ideal technique with which to probe the relative orientation of the dominant transition dipole moments in nonoriented quasi-one-dimensional materials. Finally, by generating the proper components of the nonlinear susceptibility tensor, we have demonstrated that the existence of nonparallel transition dipole moments in itself is not sufficient to violate Kleinmann symmetry.
199
4. Acknowledgement This research was supported by the US Department of Energy. We thank B.S. Hudson, D. McBranch, K. Pakbaz, A. Redondo and D. Smith for important comments and discussions.
5. References [ 1 ] T.W. Hagler, K. Pakbaz and A.J. Heeger, Phys. Rev. B, submitted for publication. [2] R. Worland, SD. Phillips, W.C. Walker and A.J. Heeger, Synth. Metals 28 ( 1989) D663. [ 3 ] S.D. Phillips, R. Worland, G. Yu, T.W. Hagler, R. Freedman, Y. Cao, V. Yoon, J. Chiang, WC. Walker and A.J. Heeger, Phys.Rev.B40(1989)9751. [4] A. Horvath, H. Bassler andG. Weiser, Phys. Stat. Sol. B 173 (1992) 755. [5]O.M. Gelsen, D.A. Halliday, D.D.C. Bradley, P.L. Bum, A.B. Holmes, H. Murata, T. Tsutsui and S. Saito, Mol. Cryst. Liquid Cryst. 216 (1992) 117. [6] C. Botta, G. Zhuo, O.M. Gelsen, D.D.C. Bradley and A. Musco, Synth. Metals 55 (1993) 85. [7]B.J.OrrandJ.F. Ward,Mol.Phys.20 (1971) 513. [ 81 R. Wortmann, P. Kramer, C. Glania, S. Lebus and N. Detzer, Chem. Phys. 173 ( 1993) 99. [9] D.A. Kleinmann, Phys. Rev. 126 (1962) 1977. [ lo] Y.R. Shen, The principals of nonlinear optics (Wiley, New York, 1984).
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 2 18 ( 1994) 195 199
Nonparallel transition dipole moments and the polarization dependence of electroabsorption in nonoriented conjugated polymer films T.W. Hagler Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 20 October 1993; in tinal form 29 November 1993
Abstract We present a comprehensive analysis of the polarization dependence of the field-induced absorption in nonoriented conjugated polymer films. The results demonstrate that the polarization anisotropy of the field-induced absorption is directly related to the relative angle, A&, between the dominant one-photon and two-photon transition dipole moments responsible for the nonlinear effect. The polarization anisotropy has a maximum value of 3 : 1 for parallel transition dipole moments, and a minimum value of 1: 3 for perpendicular transition dipole moments. We demonstrate that the predicted field-induced absorption anisotropy is consistent with Kleinmann symmetry for any relative orientation of the dominant transition dipole moments.
1. Introduction Electric field modulation spectroscopy (EFMS) has recently become an important technique for investigating the electronic structure of nonlinear organic chromophores. One aspect of EFMS, which is often overlooked, is that the technique involves the mixing of two independently polarized electric fields, i.e. the optical field and the applied dc electric field. As a consequence, one can determine two independent components of the macroscopic nonlinear susceptibility tensor by measuring the field-induced absorption or reflectance for optical polarization parallel and perpendicular to the applied dc electric field. By analyzing the ratio of these two components, one can then extract information pertaining to the microscopic electronic structure of the material. In a recently submitted article, Hagler, Pakbaz and Heeger [ 1 ] demonstrated that the polarization dependence of the field-induced absorption in oriented
conjugated polymer samples is related to the angle of the dominant G to ‘9%”transition dipole moment with respect to the drawing axis, and that in certain materials, this angle is related to the spatial extent of the excited state wavefunction. In contrast, the polarization anisotropy of the nonoriented sample, defined as the ratio of the field-induced absorption for optical polarization parallel and perpendicular to the applied dc electric field, has not been sufficiently examined. Initial investigations suggested that the polarization dependence of the field-induced absorption in nonoriented conjugated polymer films was also related to the off-axis component of the transition dipole moment [ 2,3 1. However, this analysis was based on the 22 = 4 projections of the optical and dc electric field intensities onto the polymer chain axis, rather than the 42 = 16 projections of the electric field amplitudes. By considering the various projections of the optical and dc electric field amplitudes onto the poly-
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI0009-2614(93)El486-Z
196
T. W. Hagler / Chem. Phys. Letters 218 (1994) 195-199
mer chain axis and then averaging over all possible chain orientations, one obtains an expression for the polarization dependence of the field-induced absorption that is independent of the angle the transition dipole moment makes with respect to the polymer chain axis. In fact, any theoretical model (see for instance ref. [4] ), that assumes that the dominant transition dipole moments are parallel will predict a field-induced absorption polarization anisotropy ratio of 3 : 1, regardless of the angle the transition dipole moments make with respect to the polymer chain axis. In nonoriented conjugated polymer samples, the polarization dependence of the field-induced absorption routinely violates this predicted anisotropy. In this Letter, we examined the polarization dependence of the field-induced absorption for an isotropic distribution of quasi-one-dimensional nonlinear chromophores of Cl,, spatial symmetry. Our results show that the polarization dependence of the macroscopic nonlinear susceptibility, xf.&( w, 0,O) depends critically on the details of the microscopic nonlinear polarizability, y&a (w, 0,O). In particular, the observed polarization dependence of the tield-induced absorption is directly related to the relative angle, A&, between the two dominant transition dipole moments responsible for the nonlinear effect (‘9, Ir IG) and ( “4 ( r I ‘%I,), respectively. The polarization anisotropy has a maximum value of 3 : 1 for parallel transition dipole moments, A&=0, and a minimum value of I : 3 for perpendicular transition dipole moments, A&=x/2. In addition, we calculate several components of the macroscopic nonlinear susceptibility tensor x1,&( w, 0,O) to demonstrate that the field-induced absorption anisotropy is consistent with Kleinmann symmetry for any value of the relative angle A&.
involved in the nonlinear response. To simplify the discussion, we use the geometry defined in Fig. 1 and assume that the applied dc electric field I;is along the X axis, and that the incident optical field E( k, w) is propagating in the 2 direction and is polarized at an angle of 8 with respect to the electric field. The distribution of polymer chains is assumed to be random in three dimensions; however, the plane defined by the optical polarization and the electric field polarization reduces the problem to two dimensions. The nonlinear wave equation describes the propagation and generation of electromagnetic fields in the system due to the applied dc electric field perturbation and the incident optical field, and is, therefore, the most general description of electro-optic phenomena, describing both electroabsorption and electroreflectance. In this analysis we will focus exclusively on the field-induced absorption; the corresponding expressions for the electroreflectance can be obtained by applying the appropriate boundary conditions. Using the definitions of the complex linear dielectric function, Z= 1+4x~(‘)=N& and the complex linear refractive index, No = no+ iKo,the nonlinear wave equation describing the modulation of the optical field by the applied dc electric field can be written as
E(k.a)
X, F
2. Analysis of the nonlinear response 2.1. Polarization dependence of-AT/T(B)
In this section we examine the polarization dependence of the measured nonlinear response -AT/ T( 0) with respect to the applied electric field. The objective is to demonstrate that the observed polarization dependence is related to the relative angle A&, between the two dominant transition dipole moments
Fig. 1. The geometry used to analyze the polarization dependence of the field-induced absorption. The upper-case coordinates define the axis of the macroscopic linear and nonlinear susceptibility, and in amorphous, or polycrystalline materials, are dictated by sample geometry. The lower-case coordinates define the axis of the microscopic linear and nonlinear polarizability, and refer to the axis of particular polymer chain.
T. W. Hagier /Chem. Phys. titters 218 (1994) 195-199
x(F-.l)(F-R)(&L)eiZn']E(k,o,z)=O,
(1)
where Zc,= w/c is the free space wavevector of the optical field, and $2=0 is the frequency of the applied dc electric field. In writing Eq. ( 1)) we have used the experimental observation that the electroabsorption lineshape tends to resemble a positive derivative of the unperturbed absorption coefficient. This dictates that the subscripts Z and L refer to the matrix elements associated with the optical field and J and K refer to matrix elements associated with the applied dc electric field [ 5 1. Note that the upper-case coordinates Z, J, K and L refer to the macroscopic coordinate system of the sample, and that the macroscopic linear and nonlinear polarization P(I) and PC3) are obtained by a vector summation over p(l) and c3) the microscopic linear and nonlinear polarizaP 3 tions of the individual polymer chains. Hence, in nonoriented samples, the subscripts on xf& have nothing to do with the polymer chain axis, but are dictated by sample geometry. For the sample geometry shown in Fig. 1, it is convenient to choose Z, J, K, L=-lx r>. By integrating the solution of Eq. ( 1) over the thickness of the sample, we obtain the experimentally measured nonlinear response -~(W,e)-dAOl(W,8)eizn’,
(2)
where Acy (w, .9) is defined as the polarization-dependent field-induced absorption coefficient, which can be written in terms of the macroscopic nonlinear susceptibility tensor as Aa(w,
4x 0) =k, -Im[ZCWL(o, ,N0 ,2
+ZV:#&(o,
0, 0)sin2(@]F2.
O,O)cosW (3)
Because of the isotropic distribution of the conjugated polymer chains, the linear optical coefficients are polarization independent. Hence, the polarization dependence of the measured nonlinear response, - AT/ T( t9) is contained entirely in the third-order nonlinear susceptibilityX(3’ (0, 0,0), which, in turn, depends on the details of the microscopic nonlinear
197
polarizability. Therefore, to understand the physical origin of the polarization dependence of the tield-induced absorption in a macroscopic sample, we need to consider the microscopic polarizability of the individual polymer chains. The nonlinear response of an individual chain is proportional to the third-order polarizability tensor y& (0, 0, 0 ), where the indices i, j, k, I are defined in a local coordinate system with respect to the polymer chain axis. Since there are two independently polarized electric fields, E( k, w) and F, the 8 component of the hyperpolarizability of a polymer chain oriented along the C#I axis is given by the various projections of the optical field and the applied dc electric field onto the unit vectors i, j, k, 1. As stated previously, the observed electroabsorption lineshape dictates that the subscripts i and 1refer to the matrix elements associated with the optical field and j and k refer to matrix elements associated with the applied dc electric field [ 51. As a result, there are 42 = 16 independent terms in the microscopic nonlinear polarizability which contribute to the macroscopic nonlinear susceptibility. The final expression for the macroscopic nonlinear susceptibility of the nonoriented sample is obtained by averaging over all possible chain orientations,
x~~~(w,O,O)=tl~ij~~~~y~~~(w,O,O) ,,, , x(i-4)ci’f)(M)(m))
(4)
where the lower-case subscripts x and y refer to directions parallel and perpendicular to the molecular axis of the polymer chain, respectively. Note that possible rotations about the polymer chain axis are incorporated into the averaged perpendicular coordinate y. By inserting Eq. (4) into Eq. (3) we obtain the following polarization dependence for the lield-induced absorption of the nonoriented sample:
- A+,C+&A~,,,(w){[3-2cos2(A&)] -cos2(0)[2-4cos2(A&)]),
(5)
where A@*is the relative angle between the dominant G to ‘9, and ‘B, to “‘4 transition dipole moments,
T. W. Hagler / Chem. Phys. Letters 218 (1994) 195-199
198
(6) and
where rlf)(o, 0,O) is the component of the microscopic nonlinear polarizability associated with the optical polarization parallel to the G to kB, transition dipole moment, and the electric field parallel to the ‘%, to “‘A, transition dipole moment (i.e. E(k, w) and Fare along the $A and aji axis, respectively). It is clear from Eq. (5) that if the two dominant transition dipole moments responsible for the fieldinduced absorption are parallel, A& = 0, we expect to see a polarization anisotropy of 3 : 1 in the nonlinear response. If, on the other hand, the two dominant transition dipole moments are perpendicular, A&=x/2, we expect to see an anisotropy of 1: 3 in the nonlinear response. To facilitate the analysis of experimental data, Eq. (5) can be inverted to yield the relative angle between the dominant transition dipole moments as a function of the measured anisotropy 3-P COS~(A@~)= 2+2p’ where p= AT( 90” ) /AT( 0” ). Since deviations from the 3 : 1 ratio predicted for parallel transition dipole moments are common in the literature [ 2,3,6,7], the physical origin of the nonparallel transition dipole moments becomes an essential ingredient to a more complete description of the nonlinear response. 2.2. Kleinmann
symmetry
It has recently been suggested [ 81 that the existence of low-lying perpendicular electronic bands leads to a violation of Kleinmann symmetry [ 9, lo], which predicts the following relationship between the various components of the nonlinear susceptibility tensor:
In order to investigate possible violations of Kleinmann symmetry within our model for the polarization dependence of the field-induced absorption, which includes arbitrarily nonparallel transition dipole moments, we generate the following components of the third-order nonlinear susceptibility tensor:
xBn4w
090)
= &YIF’( w,O,O)[-3+4cos2(A@~)], Xlr33rx(~, O,O) 1 = ,,rl;‘(
XGMW
0,0,0)[3-2cos2(A&],
090) = &F’cw,
0, 0) *
(10)
From Eq. ( lo), we see that our model for the polarized field-induced absorption is consistent with Kleinmann symmetry for any value of the relative angle A&.
3. Conclusions We have reexamined the analysis of the polarization dependence of the field-induced absorption for nonoriented quasi-one-dimensional samples and found a discrepancy with previously published results, which considered only the projections of the optical and dc electric field intensities onto the polymer chain axis. Our analysis, which considers the projections of the optical and dc electric field amplitudes onto the polymer chain axis, relates the observed anisotropy of the field-induced absorption in the nonoriented material to the relative angle A@, between the dominant G to 9” and ‘%, to “AB transition dipole moments responsible for the nonlinear effect. The polarization anisotropy of the field-induced absorption in nonoriented samples has a maximum value of 3 : 1 for parallel transition dipole moments, Agd=O, and 1: 3 for perpendicular transition dipole moments, A& = ~12. The results are independent of the localization length of the excited state wavefunc-
T. W. Hagler / Chem. Phys. Letters 218 (1994) 195-199
tion, or the angle that the transition dipole moments make with respect to the molecular axis. It is clear that by understanding the origin of offaxis and nonparallel transition dipole moments, one would achieve a more comprehensive picture of the nonlinear hyperpolarizability tensor, r&i], in quasione-dimensional materials such as polyacetylene and poly(pphenylenevinylene). In the absence of disorder, the relative angle between the dominant G to ‘B, and “B, to “‘A, transition dipole moments is a fundamental quantity of the material which should not depend on the parameters of a particular model. Therefore a direct measurement of A& would be a valuable asset in determining which theoretical models are best at describing a particular system. Because of its simplicity, polarized EFMS is an ideal technique with which to probe the relative orientation of the dominant transition dipole moments in nonoriented quasi-one-dimensional materials. Finally, by generating the proper components of the nonlinear susceptibility tensor, we have demonstrated that the existence of nonparallel transition dipole moments in itself is not sufficient to violate Kleinmann symmetry.
199
4. Acknowledgement This research was supported by the US Department of Energy. We thank B.S. Hudson, D. McBranch, K. Pakbaz, A. Redondo and D. Smith for important comments and discussions.
5. References [ 1 ] T.W. Hagler, K. Pakbaz and A.J. Heeger, Phys. Rev. B, submitted for publication. [2] R. Worland, SD. Phillips, W.C. Walker and A.J. Heeger, Synth. Metals 28 ( 1989) D663. [ 3 ] S.D. Phillips, R. Worland, G. Yu, T.W. Hagler, R. Freedman, Y. Cao, V. Yoon, J. Chiang, WC. Walker and A.J. Heeger, Phys.Rev.B40(1989)9751. [4] A. Horvath, H. Bassler andG. Weiser, Phys. Stat. Sol. B 173 (1992) 755. [5]O.M. Gelsen, D.A. Halliday, D.D.C. Bradley, P.L. Bum, A.B. Holmes, H. Murata, T. Tsutsui and S. Saito, Mol. Cryst. Liquid Cryst. 216 (1992) 117. [6] C. Botta, G. Zhuo, O.M. Gelsen, D.D.C. Bradley and A. Musco, Synth. Metals 55 (1993) 85. [7]B.J.OrrandJ.F. Ward,Mol.Phys.20 (1971) 513. [ 81 R. Wortmann, P. Kramer, C. Glania, S. Lebus and N. Detzer, Chem. Phys. 173 ( 1993) 99. [9] D.A. Kleinmann, Phys. Rev. 126 (1962) 1977. [ lo] Y.R. Shen, The principals of nonlinear optics (Wiley, New York, 1984).