The Kramers-Kronig relations between the real and imaginary parts of a response function are widely used in solid-state physics to evaluate the corresponding quantity if only one component is measured. They are among the most fundamental statements since only based on the analytical behavior and causal nature of the material response [Phys. Rev. 104, 1760–1770 (1956)]. Optical losses, for instance, can be obtained from the dispersion of the dielectric constant at all wavelengths, and vice versa [Handbook of optical constants of solids, Vol. 1, p. 35]. Although the general validity was never casted into doubt, it is a longstanding problem that Kramers-Kronig relations cannot simply be applied to anisotropic crystalline materials because contributions from different directions mix in a frequency-dependent way. Here we present a general method to identify frequency-independent principal polarizability directions for which the Kramers-Kronig relations are obeyed even in materials with lowest symmetry. Using generalized spectroscopic ellipsometry on a single crystal surface of triclinic pentacene, as an example, enables us to evaluate the complex dielectric constant and to compare it with band-structure calculations along the crystallo-graphic directions. A general recipe is provided how to proceed from a macroscopic measurement on a low symmetry crystal plane to the microscopic dielectric properties of the unit cell, along whose axes the Kramers-Kronig relations hold.
©2008 Optical Society of America
For centuries materials are characterized by their macroscopic optical properties, and the experimental investigations and theoretical understanding are developed to a very high degree . On this background it surprises to realize that basic questions like the complex optical behavior of low-symmetry absorbing crystals are not treated in a satisfactory way. The reasons for that are as simple as fundamental. Optical properties in the infrared, visible and ultraviolet spectral range are determined by electronic excitations, i.e. are linked to the electronic structure which can be evaluated as soon as the crystal structure is known. In band-structure calculations the dispersion is given along the high-symmetry directions of the crystal, which are defined by the unit cell [e.g. Fig. 1(a)]. Along these directions the optical transitions and polarization can be calculated, and eventually the microscopic optical functions are evaluated for the crystal axes. Only in crystals with orthorhombic or higher symmetry, the axes are perpendicular to each other and light polarized along one axis probes solely this direction. For lower-symmetry crystals, contributions from the various directions always add up in certain ways. From this superposition the real and imaginary parts of the macroscopic dielectric tensor ε̄1 and ε̄2 are obtained, respectively, and which are expressed in common Cartesian coordinates often pertaining to the exterior crystal surface under study; the principal axes of the real and imaginary parts may be differently oriented as depicted in Fig. 1(c). Most important, the dielectric tensors are not fixed with varying energy but may rotate and change shape for light of different wavelength. These dielectric tensors determine the optical response, as obtained by reflection measurements off a crystal surface, for instance.
2. Outline of the problem
Characterizing a crystal by optical measurements requires to go these two steps backwards if the intrinsic dielectric properties ought to be obtained. Although the effect is purely geometrical, it turns out to be far from trivial to disentangle the contributions in anisotropic absorbing crystals. Nevertheless, people have learnt over the centuries how to deal with these difficulties in the case of transparent materials . From optical measurements along different orientations of a crystal, one can finally extract the 3×3 dielectric tensor; this describes the value and angular dependence of the dielectric constant ε 1 . As in any ellipsoid, the three principal axes are perpendicular to each other [Fig. 1(c)], but in general do not coincide with the crystal axes. In principle a second tensor for the imaginary part ε̄2 has to be determined to account for the dielectric losses. Here the complication occurs that both acquired components are not Kramers-Kronig consistent for a given direction of measurement; except when both ellipsoids are collinear at any frequency, which only occurs for symmetries equal to or higher than orthorhombic.
The final aim is to relate the measured values and orientation of the dielectric tensors to the band structure. In general, for each photon energy we obtain three principal axes for the real part of the dielectric tensor and another three for the imaginary part; and their orientation with respect to the crystal axes is described by two times three angles, for each energy. The paramount problem originates in the fact that the microscopic optical functions calculated from band structure are linked to the crystal symmetry with non-orthogonal axes. With other words, the last step is not just a simple rotation but a decomposition of the dielectric properties. It immediately becomes obvious why this task it not trivial and has been tackled only recently. The optical behavior of a few monoclinic and triclinic crystals was measured by optical spectroscopy [6, 5] or ellipsometry [7, 8, 9]. The rotation of the principal dielectric axes with energy is discussed in all cases but prevented the authors from providing the microscopic dielectric functions along the crystallographic axes (cf. Fig. 3 and Appendix). Suitable but incorrect assumption have to be made, leading to unsatisfactory results which sometimes even violate the Kramers-Kronig relations. The problem could be solved for ellipsometry or reflection measurements off a single plane, but for a rigorous analysis, spectra at several surface orientations have to be recorded. However, the special habitus of anisotropic crystals often impedes measurements on different planes with defined orientations and in many cases it is basically impossible due to the crystal morphology.
As a matter of fact, this is not an academic problem, since the prime candidates for future organic electronics and photovoltaics, for example, the molecular semiconductors tetracene and pentacene, have triclinic symmetry. Although eventually thin films will be used for widespread applications [10, 11], any advances in their performance — which still is limited by grain boundaries, traps and imperfections — will require a deep understanding of the constituent single crystals  and a detailed knowledge of the underlying band structure. A number of band structure calculations have been published [13, 14, 15, 16], however, until now no accurate verification of the predicted electronic transitions and anisotropy have been performed by optical measurements, mainly due to the low symmetry of the crystals. In this paper we demonstrate for the first time that in triclinic crystals the intrinsic dielectric functions along the crystallographic axes can be obtained by generalized spectroscopic ellipsometry on a single crystal plane. For pentacene, as an example, we compare the experimentally received optical behavior (Fig. 3) with those obtained from band-structure calculations (Fig. 4).
3. Outline of the solution
Ellipsometry — first described and applied by Drude more than 100 years ago — is a sensitive tool to determine optical constants of bulk materials. The accuracy is considerably improved compared to simple intensity measurements as it probes only the change of the polarization of light after interaction with a sample [17, 18]. Generalized ellipsometry is the global extension to anisotropic materials [17, 19]. Here the polarization state of light and the interaction with a surface are commonly represented by the real-valued 4×1 Stokes vector and the 4×4 Mueller matrix, respectively . By measuring the change of polarization of light reflected off a surface
using sequences of differently polarized incident light, subsets or complete sets of the Mueller matrix elements are determined. For each photon energy h̄ω of the probing light, these data are measured as a function of the angle of incidence Φa, and the sample azimuth angle ϕ a, as defined in Fig. 2. From the variation of the, generally independent, Mueller matrix elements with Φa and ϕ a, the full complex-valued 3×3 dielectric function tensor ε̄ of any anisotropic and absorbing crystal is accessible .
For a proper analysis three coordinate frames must be distinguished: (i) the measurements are performed in laboratory Cartesian coordinates (x,y, z) defined by the plane of incidence (x, z) and the sample surface (x,y) as depicted in Fig. 2; (ii) the macroscopic optical response is described by the second-rank tensor of the complex dielectric function ε̄ [Fig. 1(c)]; (iii) any microscopic description is based on the electronic system which contains the crystal symmetry defined by the unit-cell vectors a, b and c [Fig. 1(a)]. The microscopic polarizability p is linked to the electronic system and thus can be calculated within this frame as a superposition of contributions along these directions
The polarizability functions ρ i may vary with photon energy. They must obey Kramers-Kronig consistency, as they descend from description of the polarization response along a given dipole in direction a, b or c.
In a reflection experiment the electric field vector E given in the laboratory system causes the macroscopic polarization P, and both are related by the second-rank dielectric tensor ε̄: P=(1-ε̄)E. To draw conclusions from measurements in laboratory frame to the microscopic electronic properties, a two-step transformation is necessary. First, the Cartesian frame is rotated by Euler angles φ,θ,ψ to an auxiliary system (ξ,η,ζ) with ζ being parallel to the crystallographic c-axis [Fig. 1(b)]. In the case of orthorhombic, tetragonal, hexagonal, and trigonal systems, φ, θ and ψ can be chosen in such way that the microscopic dielectric function is diagonal in (ξ,η,ζ). For monoclinic and triclinic systems an additional projection T onto the orthogonal auxiliary system (ξ,η,ζ) is necessary since the principal directions of the microscopic polarizability p are not perpendicular to each other . For triclinic symmetry T is given by :
Due to Eq. (1), the new angles introduced into the ellipsometric analysis should be equal to the unit cell angles α,β,γ.
Since the transformations cannot simply be inverted and an analytical expression spelled out, the task is now to obtain the best match between calculated and measured ellipsometric data, connected by a set of highly-consistent numerical functions. From the experimental side, the projection T is applied to all sets of ε̄, obtained independently for each frequency. From the microscopic side, the polarizability functions ρ a,ρ b,ρ c are varied for each frequency. In addition, the external angles φ,θ,ψ and internal angles α,β,γ (both sets constant with frequency) have to be optimized. The microscopic dielectric functions ε j=1+ρ j(j=a,b,c) are hence-forth extracted on a point-by-point basis, i.e. without any lineshape implementation, allowing for independent Kramers-Kronig consistency tests.
4. Experimental details
We have applied this analysis to obtain the intrinsic dielectric properties of pentacene from generalized spectroscopic ellipsometry measurements on a single surface. Pentacene single crystals were grown by sublimation of purified material. The crystallization took place under streaming inert gas with a hydrogen content of 4 % and a flow rate of 50 cm3/min. At sublimation temperatures of 300°C the glass oven provides a temperature gradient, which is sufficiently steep for low-defect crystallization. The crystals form flat shapes with lateral dimensions of 5×5 mm2 and thickness up to 30 µm. Sample surface oxidation was prevented by handling specimen under yellow light. Light exposure during measurement was minimized, and no time-dependent changes of the measured data were observed. The parameters of the pentacene triclinic unit cell [Fig. 1(a)] obtained by x-ray diffraction on the crystals investigated here are in good agreement with literature values [26, 27, 28].
The crystals were mounted with the (ab) plane normal to the axis of the ϕ a rotation (Fig. 2). The angle between the (ab)-plane and the c-axis is 12.2°. The a-direction was aligned along the plane-of-incidence at ϕ a=0°. The external angles φ, θ, and ψ given by our sample mount procedure were confirmed by the elllipsometric measurements. For the angles between the principal polarizability directions we found α=95.4°, β=101.4°, and γ=95.3° in excellent agreement with the unit-cell angles obtained from our x-ray diffraction investigations on the same crystal. This consistency provides first clear evidence that there exist frequency-independent principal polarizibility directions, which coincide with the unit-cell directions, and therefore built the natural frame to describe the optical properties of low-symmetry crystals.
Ellipsometric measurements were performed with a micro-focus, rotating-compensator-type multichannel ellipsometer system (M-2000™, J. A. Woollam Co., Inc.) mounted to a goniometry stage. Data were taken in the spectral range between 370 nm (photon energy 3.35 eV) and 990 nm (1.25 eV) at a resolution of about 14 meV. The angle of incidence Φa was varied between 45° to 75° at a step width of 5°; the azimuth angle ϕa from 0° t360° in 22.5° steps (cf. Fig. 2). Normalized Mueller matrix elements were recorded for each set of (Φa,ϕ a,ω). For each frequency ω, the generalized ellipsometric data allow for reconstruction of ε̄, as shown previously for orthorhombic symmetry materials .
Fig. 3 depicts the dielectric functions ε 1 and ε 2 of triclinic pentacene along the crystallographic axes a, b, c. The response along axis a can be described by four Gaussian oscillators, three oscillators are needed along b and c, respectively; their parameters are listed in Table I. A nearly perfect match is obtained for both ε 1 and ε 2 using the same set of oscillators, which proves that the dielectric functions obtained for each axis satisfy the Kramers-Kronig relations. This is a non-trivial result, because it demonstrates for the first time that by choosing the correct basis for the microscopic polarizability directions, a self-consistent description of the macroscopic optical properties of low-symmetry crystals can be achieved!
Our findings can now be used to get more insight into the electronic properties of the molecular semiconductor. In an isolated pentacene molecule the lowest electronic excitation occurs at 2.31 eV . This π-π*-transition is polarized along the short axis of the molecular plane. In the solid the intermolecular interaction leads to the formation of bands and to the appearance of excitons. For crystals with more than one molecule per unit cell basis, the interaction of translationally inequivalent molecules causes Davydov splitting  of the excitonic states. These states have not only different energies but also different polarizations, which are characteristic for the symmetry of the crystal. For triclinic pentacene with two molecules per unit cell, the two lowest-energy excitons are polarized along the a and b directions, respectively. The excitonic transition observed here at 1.82 eV is indeed polarized along the crystallographic a axis only, where the spatial overlap to adjacent molecules is largest. The exciton peak along b occurs at 1.97 eV and is considerably damped. Due to the small overlap of the π -electrons along c their polarizability is small.
A comparison of our findings with ab-initio calculations by Tiago et al. , reproduced in Fig. 4, gives some agreement but also reveals significant discrepancies. They predict a charge-transfer exciton — smeared out over many unit cells along the (ab) plane and confined within one lattice constant along c direction — with a Davydov-splitting of 130 meV, but with opposite polarization, i.e. the lowest excitation is calculated along b while it is detected along the shortest direction a. The intensity of the strongest exciton peak is calculated correctly, however, the amplitude of the second Davydov component is not strongly reduced, but shows up in experiment with nearly the same height of ε 2≈4 [cf. Figs. 3 and 4(b)]. The measured width of the two lowest excitations is influenced by electron-phonon coupling and inhomogeneous broadening, factors not included in the calculations.
The second type of excitations occurs in the range 2…3 eV and is consentiently attributed to interband transitions between the highest occupied and lowest unoccupied bands. In our measurements ε 2(ω) is dominated in this range by one broad Gaussian oscillator for each direction centered at 2.10 eV (a) and 2.18 eV (b), respectively. Despite these concurrences, the fine structure of the calculated and measured dielectric functions between 1.5 and 2.5 eV is very different. The main reason might be that the model neglects electron-phonon coupling as well as temperature dependent effects. For example, the strong peak observed at 2.12 eV for E ‖ b has no correspondence in the calculations. It is of vibronic origin  and is assigned to a C-H in-plane bending mode strongly coupled to the exciton. We can even identify a small signature of the second vibronic replica in the measured data at 2.27 eV.
This example proves the power of generalized spectroscopic ellipsometry to obtain the complex dielectric functions along the principal polarization axes even in triclinic systems. The optically determined principal polarization axes are frequency independent and in perfect agreement with the crystallographic directions. Along these directions the complex dielectric function ε̄ of a triclinic crystal obeys the Kramers-Kronig relations. Choosing this microscopic frame of reference instead of the macroscopic polarization axes enables a quantitative comparison possible of optical experiments with band-structure calculations even in the case of low-symmetry crystals.
If the crystal symmetry is approximated to be orthorhombic, the dielectric tensors can be calculated for ellipsometric measurements on a (ab) plane. In this case the principal axes of the dielectric tensors do not depend on frequency. With other words, the projection T is not performed, leading to incorrect results . Fig. 5 compares the spectral dependence of ε 1(ω) and ε 2(ω) along the three principal aces x, y, and z with the dielectric functions along the crystallo-graphic axes a, b, and c. It becomes obvious that the absolute values are incorrect. While for x and y the main feature can be recognized, for the perpendicular direction z unreal peaks show up and others are missing; in this direction the absolute value is off by an order of magnitude.
We thank Tino Hofmann and Wolfgang Frey for fruitful discussions. The work at Stuttgart was supported by the Deutsche Forschungsgemeinschaft (DFG) under GO 642/6 and PF 385/4. M.S. acknowledges startup funds from CoE and J.A.Woollam Foundation, and support by NSF in MRSEC QSPIN at UNL.
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28. For comparison the parameters given in Ref.  and Ref.  are transformed into the parameters of the Niggli cell by using the matrices (-1 0 0/0 1 0/-1 0 -1) and (0 1 0/1 0 0/??? 0 -1), respectively.