J. Phys. Chem. B 2009, 113, 14165–14171
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Anisotropic Dielectric Tensor for Chiral Polyfluorene at Optical Frequencies Girish Lakhwani, Rene´ A. J. Janssen, and Stefan C. J. Meskers* Molecular Materials and Nanosystems, EindhoVen UniVersity of Technology, P.O. Box 513, 5600 MB EindhoVen, The Netherlands ReceiVed: July 30, 2009; ReVised Manuscript ReceiVed: September 10, 2009
The anisotropic dielectric tensor of chiral poly[9,9-bis((3S)-3,7-dimethyloctyl)-2,7-fluorene] is determined via variable angle spectroscopic ellipsometry and circular dichroism spectroscopy. The uniaxial anisotropy indicates a high in-plane alignment of polymer chains in the thin film. Chirality of the polymer results in small but nonzero off-diagonal matrix elements in the dielectric tensor, indicating a helical interchain organization in the vertical direction. This method for determining the dielectric tensor appears to be selfconsistent when checked against various theoretical models for the optical activity in the reflection of light. The dielectric constants for the chiral polyfluorene are compared to those for an achiral polyfluorene. 1. Introduction Conjugated polymers such as polyfluorene and its derivatives can be applied as thin layers in a variety of optical1 and optoelectronic devices.2 To understand and describe device performance and phenomena, knowledge of the optical properties of the polymer layer is of importance. The response of any material to alternating electromagnetic fields can be described by the frequency dependent dielectric constant ε(ω) ) ε1(ω) + iε2(ω), where ε1 and ε2 are the real and imaginary parts and ω is the angular frequency of the wave. Optical properties such as the complex refractive index N(ω) ) n(ω) + ik(ω), in which n is the index of refraction and k the extinction index, are directly related to the dielectric constant via N2 ) ε. Spin-coated films of polyfluorene and other conjugated polymers often exhibit anisotropy.3-7 This phenomenon can be related to alignment of chains during the deposition process. In some cases, this alignment of chains has been supported by X-ray diffraction measurements.8 Polyfluorenes have a rigid rodlike character and also display anisotropy in their optical properties at the molecular level. For instance, the transition dipole moment associated with the lowest optical transition is oriented along the chain direction. The combination of preferential alignment of the chains and the directionality of the molecular optical transitions results in anisotropic optical properties of the film. Spectroscopic ellipsometry can be employed to determine anisotropic optical constants in thin films of polyfluorene.9-12 Knowledge of this optical anisotropy is important for modeling the performance of optoelectronic devices. For polymer photovoltaic cells, the absorbance of sunlight may be increased by aligning the polymer chains in the plane of the film.13 Likewise, out-coupling of light out of a polymer light emitting diode is influenced by the anisotropy.14 The optical anisotropy can be uniaxial or biaxial in nature. For the latter, the refractive index is different in all three directions, whereas in the former, the index assumes identical values along two principal axes. For uniaxial anisotropy, the frequency dependent dielectric tensor can be represented as follows: * E-mail:
[email protected].
ε(ω) ) ε0
[
ε⊥(ω) 0 0 ε⊥(ω) 0 0 (ω) ε 0 0 |
]
(1)
where ε⊥ and ε| are components of the dielectric constant of the medium for light with an electric field perpendicular and parallel to the optical axis and ε0 is the vacuum permittivity. The dielectric tensor describes the polarization of the material under the influence of the applied field E0. Often it is assumed that the polarization P(r) at the position r depends only on the value of the externally applied field at position r (dipole approximation). However, this is not necessarily true and the polarization may also depend in a nonlocal manner on the applied field. The nonlocality of the response, often also referred to as spatial dispersion, can be incorporated into the dielectric function by including a dependence on the wavevector k of the light. In a series expansion, one can write15
ε(ω, k) ) ε(ω) + iγk + Rkk + ...
(2)
where γ is a material’s constant and k is the wavevector (|k| ) ωn/c, with n as the refractive index). For centrosymmetric materials, γ ) 0 and the first k dependent term in eq 2 is quadratic. Here we assume that all quadratic terms are negligible. For chiral materials, γ can have nonzero values that describe the natural optical activity. Far away from any optical resonance, the value of γ relates to the magnitude of the optical rotation. The dielectric tensor for chiral materials with uniaxial anisotropy can now be written as
ε(ω) ) ε0
[
]
ε⊥(ω) 0 0 ε⊥(ω) 0 + 0 ε|(ω) 0 0 -Gxy(ω) -Gxz(ω) 0 -Gyz(ω) i Gxy(ω) 0 Gxz(ω) Gyz(ω) 0
[
]
(3)
where the second term on the right-hand side, G ) γk is a pseudotensor that changes its sign upon the inversion symmetry operation.
10.1021/jp907297y CCC: $40.75 2009 American Chemical Society Published on Web 09/28/2009
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Figure 1. (1) Poly[9,9-bis((3S)-3,7-dimethyloctyl)-2,7-fluorene] and (2) poly[9,9-di-n-octyl-2,7-fluorene].
A light wave propagating along the optical axis (z) through the dielectric material obeys the following wave equation:
[ ] ∂2 ∂x2
[] [
0
2
0
∂ ∂y2
][ ]
Ex 1 ε⊥ -iGxy Ex ) 2 Ey ε⊥ Ey c iGxy
(4)
The two eigen polarizations of the medium, i.e. the two polarizations that travel through the medium without changing, can be obtained from diagonalizing the relevant part of the dielectric tensor.
[
det
]
ε⊥ - λ -iGxy )0 iGxy ε⊥ - λ
(5)
Here because of the symmetry of the matrix, the eigen polarizations are purely circular. The nonzero value of Gxy describes the difference of the material in its response to left and right circular polarized light incident along the optical axis. The eigen values correspond to the scalar values of the dielectric constant for the left and right circular polarized eigen polarizations
λ( ) λL/R ) εL/R ) ε⊥ ( Gxy
(6)
The refractive index for the two eigen polarizations can be obtained by taking the square root of dielectric constants using Taylor series and neglecting high-order terms in Gxy
NL/R ) √εL/R = √ε⊥ (
Gxy Gxy ) N⊥ ( 2 2
(7)
The real and imaginary parts of these refractive indices NL/R ) nL/R + ikL/R can now be expressed in terms of n⊥ and the complex pseudotensor Gxy ) G1,xy + iG2,xy
kL/R ) k⊥ (
G2,xy 2
(8)
nL/R ) n⊥ (
G1,xy 2
(9)
In this study, we determine the anisotropic dielectric tensor including terms first order in k for films of a chiral polyfluorene 1 (Figure 1). The ultimate aim of this study is to understand the relation between optical properties of the material and the molecular organization of the polymer chains in the film. For achiral polyfluorenes, ε⊥(ω) and ε|(ω) have been studied, yet for chiral polyfluorene, no data for ε⊥(ω) and ε|(ω) are
available. Here, we use spectroscopic ellipsometry to determine these quantities. For films of chiral polymer, the tensor elements G also contain information on the molecular organization in the film. In this study, our aim is to determine a value for G and to interpret these in terms of molecular organization. Here we encounter the difficulty that no standardized procedure exist to determine this quantity. Traditionally, G is determined from transmission measurements. However, when studying solids instead of dilute solutions, the characteristics of the transmitted light are influenced by both its interaction with the bulk and the reflection of the light at the surface. Therefore, ideally, one would like to obtain values for G from reflection measurements. Yet, for reflection of left and right circular polarized light from the surface of chiral materials, the differences in the amplitude and phase shift of the reflected beams with opposite circular polarization are not well understood. There are reports claiming the existence of such optical activity in reflection of light incident along the surface normal of a chiral dielectric,16 while other studies have denied the existence of such effects for homogeneous, enantiomorphous materials for light under normal incidence17,18 or failed to detect any such effect.19 In view of this controversy, determination of G from reflection measurements alone is practically not feasible. Up to now, only few studies20 have attempted to determine chiroptical parameters from reflection measurements using recently developed matrix methods to describe the effects.21 Here, we follow a different strategy. We determine G(ω) from transmission measurements monitoring the difference in absorption of left and right circular polarized light (circular dichroism, CD). We then use existing theory for circular selectivity in the reflection of light in order to estimate to what extent the CD might be influenced by the selective reflection. The pseudotensor elements G are also known to influence the reflection of light from the surface of a chiral material for light coming in under an angle.17 Using the values for G from CD measurements, we will then asses to what extent the ellipsometric determination of ε⊥(ω) and ε|(ω) might be affected by the chirality dependent term in ε. In this way, we arrive at a self-consistent estimate of ε⊥(ω), ε|(ω), and G(ω). Here we focus on thin films of 1 obtained by spin coating from chloroform solution and without any further thermal treatment. It has been shown previously22 that chiroptical properties for unannealed films are intensive in nature, i.e., G independent of the film thickness. This is important, because the dielectric tensor ε(ω) in eq 3 is by definition an intensive property. It is well-known that molecular materials with longrange helical order, e.g. cholesteric liquid crystals, show selectivity in the reflection of left and right polarized light and that this selectivity depends on the film thickness.23 This selectivity is therefore not an intensive optical property of a thin film, and a dielectric tensor such as in eq 3 cannot be used to describe the optical behavior of these chiral materials. Also thermally annealed films of polyfluorene have been found to display thickness dependent chiroptical properties.24 Thus, in order to avoid these complications in dielectric behavior arising from long-range order, we study here unannealed films, for which the chiroptical properties are known to be intensive. Finally, both the chirality dependent terms and the anisotropy of the dielectric response of the film of chiral polyfluorene 1 will be interpreted in terms of the molecular orientation and organization in the film.
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Figure 2. (a) Ψ and (b) ∆ values recorded by the ellipsometer for the thin film of 1 on an Si wafer of 65 nm thickness at different angles of incidence. Also shown are the fits of the anisotrpic model to the data.
Figure 3. Anisotropic optical constant (n and k) values of a thin film of 1 on an Si wafer as derived from the model fit for films with two different thicknesses (38 and 65 nm).
2. Experimental and Modeling Section 2.1. Film Preparation and Instrumentation. Chiral polyfluorene poly[9,9-bis((3S)-3,7-dimethyloctyl)-2,7-fluorene] (1, Figure 1, Mw ∼ 37 600, polydispersity index (PDI) 1.84) films were prepared by spin coating the chloroform solution (conc. 10 mg/mL) on Si wafers (with a similar copy on glass substrates) at a spin rate of 1500 rpm. The synthesis of 1 has been described elsewhere.25 The films obtained were homogeneous with a thickness of approximately 65 nm as recorded using a Veeco surface profilometer. By varying the spin rate, films of varying thickness were obtained. For achiral polyfluorene poly[9,9-din-octyl-2,7-fluorene] (2, Figure 1, ADS129BE, American Dye Source Inc., Mw ∼ 42 000), an 80 nm thick film was prepared by spin coating from a 12 mg/mL solution. Ellipsometric experiments were done on a VASE instrument (J. A. Woollam Co., Inc.). UV-vis measurements were done on a Perkin-Elmer Lambda 900 UV/vis/NIR spectrometer. CD spectra were measured on a Jasco J-815 spectropolarimeter where the sensitivity and scan rate were chosen appropriately. 2.2. Ellipsometry Experiments and Modeling. The ellipsometer yields Ψ and ∆ values, which are measures of respectively the relative attenuation of the intensity of the s and p polarized components of the light upon reflection and the corresponding relative phase angle change upon reflection. To obtain optical constants n and k, a model is used starting with an approximate input solution to the experimental data. The model contains mathematical descriptions of the complex dielectric constant of the various layers and information on the thickness and roughness of each individual layer in a stack.
Using the Levenberg-Marquardt algorithm, a curve is then fitted to the experimental Ψ and ∆ values. To obtain an optimum solution, several iterations are performed over the various fitting parameters in order to minimize the difference between experimental and modeled parameters. The fitting parameters are the thickness of the layers and the parameters that describe the complex dielectric functions in terms of dispersion equations and oscillators. More information on types of oscillators can be found in the Supporting Information. 3. Results and Discussion 3.1. Anisotropic Optical Constants of Chiral Polyfluorene with Thickness and Substrate Variation. In order to determine anisotropic optical constants for thin films of polyfluorene 1 on an Si wafer, values for the ellipsometric angles Ψ and ∆ were determined at several angles of incidence, both above and below the Brewster angle (Figure 2). By fitting an oscillator model to the experimental Ψ and ∆ values, anisotropic optical constants were obtained (Figure 3). The anisotropy was found to be uniaxial in nature with an axis of anisotropy perpendicular to the plane of the substrate. Several anisotropic models were developed in order to establish the uniqueness of the fit. The optical constants reported in Figure 3 are the solution to the model with minimum MSE and correlation. More information can be found in the Supporting Information. For wavelengths near 390 nm, the extinction index k shows a maximum. This maximum corresponds to the optical transition from the ground state (S0) to the lowest excited singlet state (S1). Near the maximum, we find that the absorption index for
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Figure 4. (a) Transmission data for a 38 nm thick film of 1 on glass recorded at normal incidence with the ellipsometer, along with predictions from the isotropic and anisotropic models. (b) Transmission measurements on recorded on a UV-vis spectrometer for a 65 nm thick film of 1 at normal incidence and prediction for the absorbance from the anisotropic modeled. No corrections for incoherent scattering have been included.
the light with E in the direction of the surface normal (k⊥) is 2 times less than for light with E in the plane of the film (k|). Because the transition dipole moment for this transition (S0 f S1) lies parallel to the backbone of the polymer chain, the anisotropy indicates that the chains are preferentially oriented in an in-plane direction. In order to validate the optical constants obtained, several checks were performed. The anisotropic optical constants for thin film of 1 were determined for two polymer layers with different thickness (38.1 ( 0.5 and 65.0 ( 0.1 nm, as determined from ellipsometry, Figure 3). We find no significant dependence of the optical constants on film thickness. The thicknesses of the film determined from ellipsometry match well with those determined from surface profilometry (38 ( 2 and 65 ( 2 nm). The anisotropic model used gives a significantly better description of the data than the corresponding isotropic model (mean square error 13 vs 40, see the Supporting Information). The optical constants were also determined also on a glass substrate and compared to those on Si (see the Supporting Information, Figure S4). The ordinary indices are virtually the same; the extraordinary components show some variation with the substrate chosen. Also here, the thickness of a thin film of 1 on glass determined from the ellipsometry to be 67.8 ( 0.9 nm matches that obtained from profilometry (64 ( 1 nm). Using glass substrates, the ellipsometer can also be used to measure the relative intensity of transmitted light and these measured values can be compared with those calculated from the optical model. The transmission coefficient calculated from the anisotropic optical constants (Figure 3), and the experimental data are plotted in Figure 4a for various wavelengths. As can be seen, the traces match quite well, and the anisotropic model gives better agreement with the experimental values than the corresponding isotropic model. Finally, using the optical constants (κ) and thickness l as obtained from the model, an absorbance curve can be modeled (A ) 4πkl/(λ ln 10)). The modeled absorbance curve matches well with the experimental UV-vis spectra, thereby validating the optical constants (Figure 4b). 3.2. Comparison of Anisotropic Optical Constants of Chiral and Achiral Polyfluorene. The optical constants of achiral polyfluorene (2) were also determined via ellipsometry to allow comparison to 1. The anisotropic optical constants determined for a thin film of 2 (Figure 5) match well with the one reported in the literature.12 Details of the modeling are listed in the Supporting Information. The thickness measured via profilometry (80 ( 1 nm) matches well with the thickness
Figure 5. (a) Optical constants (n and k) for the thin film of 2 on an Si wafer.
TABLE 1: Oscillator Strengths from Dielectric Function for Thin Films of 1 and 2 on Si and Polymer Orientation Distribution Parameter dπθ oscillator strength 4eV/p ∫2eV/p ε2(ω) dω component 4eV/p ε2,⊥(ω) dω + total 2∫2eV/p 4eV/p ∫2eV/p ε2,||(ω) dω 4eV/p ordinary ∫2eV/p ε2,⊥(ω) dω 4eV/p extraordinary ∫2eV/p ε2,||(ω)
dπθ
dω
achiral 2
chiral 1
2.41
2.22
1.12 0.17 0.93
0.99 0.24 0.89
determined from ellipsometry (81.76 ( 0.07 nm). Compared to its chiral analogue 1, the ordinary extinction index k| for 2 at its maximum near 390 nm is somewhat higher. In contrast, the extraordinary index is lower for the achiral compared to the chiral material. To compare these results in a more systematic manner, we have taken the dielectric constants from the data analysis and calculated the total oscillator strength by integrating the imaginary part of ε (i.e. taking ∫ε2(ω) dω) in the region corresponding to the S0-S1 transition between 2 and 4 eV. Results are listed in Table 1. The total oscillator strength, i.e., the sum of the oscillator strength in all three direction of the material, is 9% higher for the achiral compared to the chiral material. Considering the structure of the monomeric unit of both polymers (Figure 1), we note that the chromophoric unit is the same in both materials but that the optically inert side chains occupy a larger volume fraction in the chiral material.
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Figure 6. Absorbance and CD spectra of thin chiral polyfluorene film on glass substrate (∼65 nm).
On the basis of the chemical formulas for the monomeric units (C29H40 vs C33H48) and assuming a volume occupied by the monomeric unit proportional to the number of C atoms, one expects the total oscillator strength for the achiral material to be 14% higher. Using the oscillator strengths determined for the ordinary and extraordinary directions, a polymer orientation distribution, dπθ, can be defined as4
1
dπθ ) 1+
∫ ε2,|(ω) dω 2 ∫ ε2,⊥(ω) dω
(10)
where dπθ ) 2/3 for an isotropic orientation distribution of polymer chains in the film and dπθ ) 1 for perfect in-plane orientation of the chains. The subscript θ refers to the average polar angle between the director of the polymer chains and the normal of the film. Here, we have assumed that the transition dipole for the π-π* absorption of the polymer is oriented along the polymer chain. For a thin film of 1, the value obtained dπθ is 0.89, whereas for thin film of 2 it is found to be 0.93 (Table 1). These numbers are comparable in magnitude to values reported elsewhere on various π-conjugated polymers.4,5,26 The results indicate a higher degree of orientation for the achiral polyfluorene polymer in comparison with the chiral. 3.3. Contribution of the Pseudotensor G to the Dielectric Tensor of Chiral Polyfluorene. Chiral materials show circular dichroism (CD), i.e. a difference in absorbance for left and right circularly polarized light (∆A ) AL - AR). In Figure 6, this CD is illustrated together with the absorbance for a thin, unannealed film of 1 with a thickness of 65 nm. The measurement illustrated in Figure 6 pertains to light incident along the normal of the film. The CD measured in this way is related to the Gxy element of the pseudotensor G in eq 3. Using eq 8, we can express this relation as
G2,xy(λ) ) ∆k(λ) ) kL(λ) - kR(λ) ) ln 10∆A(λ)
λ 4πl (11)
where λ is the wavelength of the light in vacuum and l is the thickness of the film. The real part G1,xy can be approximated by taking the Kramers-Kronig (KK) transform of G2,xy. This
was done by fitting two Gaussian curves, one with positive and one with negative amplitude to G2,xy. G1,xy can then be constructed as the sum of the two KK transformed Gaussian curves. In this procedure, the contribution of CD bands to the optical rotation in G2,xy outside the window of measurements is not included. This contribution will be a slowly varying offset. In the neighborhood of resonance, the calculated G1,xy should be a reasonable approximation. Results are shown in Figure 7b. Figure 7 illustrates the photon energy dependence of various elements of the dielectric tensor of chiral polyfluorene. As can be seen, the absolute magnitude of the real and imaginary parts of the element Gxy is of the order 10-4, i.e. much smaller than ε⊥. This large difference in magnitudes provides a justification for the procedure involved in determining ε⊥ and ε| from the ellipsometric data where we have neglected any chirality dependent terms in the analysis. As mentioned in the introduction, the influence of the chirality dependent pseudotensor G on the reflectivity of chiral materials is presently poorly understood. Zheludev et al.16 have proposed a relation between G and optical activity in reflection of light on the interface between vacuum (air) and an infinitely thick layer of homogeneous chiral material:
δηr + iδRr )
Gxy 1 - ε⊥
(12)
Here δηr is the degree of ellipticity (in radians) of light reflected from the interface between air and chiral dielectric material when linear polarized light is incident along the surface normal, whereas δRr (in radians) is the corresponding optical rotation of the plane of polarization in the reflected light. Using the elements of the dielectric tensor for the chiral polyfluorene (Figure 7), we can now make predictions for the specular optical activity. These predictions are shown in Figure 8a. On the basis of eq 12, rotation of linearly polarized light is predicted up to angle of 0.05 mrad (∼3 mdeg). Although these values are small, they are above the detection limit of modern optical equipment. The values predicted here are similar to those reported for R-HgS.16 For photon energies around 3.1 eV, the reflectivity of an infinitely thick layer of chiral polyfluorene reaches a maximum (Figure 8b). From the predicted ellipticity of the reflected light, we can calculate the degree of circular polarization in the reflected light grefl (see Figure 8b). We predict grefl ) 0.2 × 10-3 at 3.5 eV. This is slightly above to the detection limit for experimental methods to determine the degree of circular polarization of light by e.g. photon counting. Thus, the predictions for the specular optical activity using eq 12 seem amenable to experimental verification. Finally, we also estimate the contribution of selective reflection of circular polarized light to the measured circular dichroism. For an infinitely thick layer, we calculate a contribution as show in Figure 8c. This prediction can be compared to the circular dichroism measured for a film of 65 nm thickness. For photon energies of 3.2 eV, we estimate contribution of reflection to the measured ∆A of about 5 × 10-6 (Figure 8c). This contribution is about 10 times smaller than the experimental ∆A from the 65 nm thick film. The fact that the predicted contribution of reflection to the total circular differential transmission is significantly smaller that the experimental data used to calculate Gxy indicates that our method for determining Gxy is close to self-consistent in the case that eq 12 holds. As mentioned in the introduction, other theoretical analyses suggest that specular optical activity vanishes for
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Figure 7. (a) In-plane and out-of-plane dielectric constants of chiral polyfluorene film on glass substrate (∼65 nm). (b) Anisotropic tensor G1,xy and G2,xy of the same.
SCHEME 1: Pictorial Representation of Polymer Alignment in the Plane within a Fibrila
a
Only one chain is represented in order to maintain the clarity of the image. The dotted line in the top view image is a representation of chains twisted in the plane of the film.
Figure 8. (a) Prediction based on eq 12 for the ellipticity and angle of rotation of the plane of polarization (in 10-6 rad) after reflection of a linearly polarized light beam incident along the surface normal to an infinitely thick layer of chiral polyfluorene. (b) Corresponding predicted degree of circular polarization after reflection (grefl) and the reflectivity of the film as calculated from N (Figure 3). (c) Estimated contribution of the circular selective reflection to the measured circular differential extinction (∆A). For comparison, the CD recorded on a 65 nm thick film is also shown (solid line).
normal incidence. If theses predictions hold, then our method for estimating Gxy is free from a systematic artifact due to selective reflection. We note however that the same analyses predict circular differential reflectance light coming in under an angle.17 This would imply that our determination of the optical constant using variable angle ellipsometry might be affected by chirality dependent terms in the reflectivity. However, the theoretical analyses predict the chirality dependent terms to be on the order of G. Since in our case G is very small, we conclude that our procedure for determining the elements of the dielectric tensor of 1 yields self-consistent results that might be used in further studies to experimentally test various theories for specular optical activity by means of direct measurements circular differential reflectance. Finally, we return to the elements of the dielectric tensor of the chiral polyfluorene as shown in Figure 7. A possible
Figure 9. (a) AFM phase image of unannealed film spin coated from very dilute solution of 1 in chloroform and (b) corresponding AFM height image (height scale 20 nm, film thickness e15 nm “patchy”) of the same. The length scale of the images is 200 nm × 200 nm.
qualitative interpretation of these elements in terms of the orientation of the polymer chains in the film is illustrated in Scheme 1. We assume here that all polymer chains are aligned in the plane of the film and that there is a correlation in the orientation of the polymer chains in adjacent sublayer of the film. This gives rise to a helical arrangement of the chains through the sublayers when viewed from above. Coupling between transition dipole moments of the chains in neighboring sublayers gives rise to CD and hence to a nonzero value of Gxy. A small value of the helical twist angle between plans is compatible with small values of Gxy. Assuming that all polymers are oriented in the same plane, the elements Gxz and Gyz would be zero. Atomic force microscopy (AFM) studies on very thin films (
