The elastic constants of rubrene determined by Brillouin scattering and density functional theory

Rubrene is an organic semiconductor with a high carrier mobility that has gained much attention due to its promise for applications in field-effect transistors and flexible organic opto-electronic devices.^1–9 These device applications, however, require an understanding of the material mechanical properties, which have only been minimally studied to date.¹⁰ Brillouin spectroscopy is a non-contact, laser-light scattering technique for the measurement of elastic properties.^11,12 The measurement is at hypersonic frequencies and probes the adiabatic stress/strain response uncontaminated by plastic or viscous relaxation, providing the purely elastic stiffness tensor. Herein, we use Brillouin scattering to probe the elastic stiffness tensor, Young's moduli, and Bulk modulus of rubrene. As a complementary investigation, we also perform first principles van der Waals-Density functional theory (vdW-DFT) calculations to compute the elastic properties of rubrene. In general, the measured and computed values of the elastic constants, the reduced constants, and Young's moduli are in good agreement, providing two independent checks of material properties. The information gleaned from such studies is essential for fundamental characterization, processing, determining performance limitations, and manufacture of organic semiconductor crystals for flexible opto-electronic applications.^13,14

High quality, single-crystal rubrene (Fig. 1) was grown through the physical vapor transport (PVT) method.¹⁵ While crystals with different morphologies (lath-like, needle-like, and platelets) can be produced, lath-like rubrene crystals (Fig. 1(b)) of ∼1 to 2 cm in length, ∼3 to 5 mm in width, and ∼3 to 6 μm in thickness were used for these experiments. Single crystal X-ray diffraction (XRD) using a Bruker D8 venture with a Mo $Kα$ source (λ = 0.71073 $Å$⁠) was measured on a single crystal to identify the crystal orientation and lattice constants. The crystals used in these measurements were flat, with large (100) faces and the [001] crystal direction aligned with their long axes. Brillouin spectra were acquired using a scanning, tandem Fabry-Perot interferometer (Sandercock) with a mirror spacing of 3.5 mm (free spectral range FSR = 80.5 GHz) to 4 mm (FSR = 70.44 GHz) and with a scan rate of 500. An entrance pinhole of 450 μm was used to achieve an optimal finesse of ∼90 (mirror spacing of 4 mm) to 120 (mirror spacing of 3.5 mm) and an exit pinhole of 750 μm. A neodymium yttrium vanadate laser (λ = 532.15 nm) was used as the laser source along with a 20× Mitutoyo objective. Rubrene absorbs strongly in the green region, limiting probe geometries for these measurements to 180° backscattering. Backscattering Brillouin spectra were acquired along each crystallographic facet and edge for full determination of the stiffness tensor. The sample was suspended by touching a small piece of Scotch brand double-sided adhesive tape onto an edge of the crystal that was then placed on a stage that left the crystal suspended in free space by 3–5 mm. The adhesive tape showed no signs of chemical reactivity with the rubrene, but the material in contact with the tape was not measured. This was then mounted onto a combination of translational and rotational stages that allow the translation of the sample along three directions and rotation about two axes. For measurements along the large flat growth plane (100), we define Θ as the polar angle of the laser relative to the [100] or a axis and α as the azimuthal angle defined such that α = 0 when the laser lies in the ab plane.

Density functional theory (DFT) calculations were performed using the Vienna Ab Initio Simulation Package (VASP).^16,17 The projector-augmented wave method^18,19 was used to represent core and valence electrons. As rubrene is a molecular crystal within which intermolecular bonding is dominated by van der Waals (vdW) forces, standard DFT exchange-correlation (XC) functionals may not accurately capture the ground-state (strain-free) lattice parameters. Hence, we employed the nonlocal, van der Waals optB86b-vdW functional of Klimes et al.,²⁰ which produces relaxed lattice parameters that are in good agreement with the experiment (Table I). From convergence tests, we established a kinetic energy cutoff of 600 eV and a $4×2×2$ Monkhorst-Pack k-point mesh for Brillouin zone sampling.²¹ A Gaussian smearing of 0.05 eV was used to accelerate electronic convergence. All lattice vectors and atomic positions were relaxed using the conjugate-gradient method with a stress tolerance of 1 kbar and a force tolerance of 0.01 eV/Å. To reduce the computational cost, our DFT calculations are performed on the Niggli reduced monoclinic unit cell. Relaxed atomic positions for the monoclinic primitive cell are provided in the supplementary material. The elastic moduli, c_ijkl, were calculated by subjecting the ground-state structure to normal and shear strains within a range of $±0.75%$ (in steps of 0.25%) and fitting the resulting elastic energy (E) versus strain (ϵ_ij) data up to second order in strain. Note that at each level of macroscopic strain, only the relevant lattice vectors are held fixed, while all internal degrees of freedom (atomic positions) are self-consistently relaxed.

Rubrene has an orthorhombic crystal structure with lattice parameters a = 26.965 $Å$⁠, b = 7.206 $Å$⁠, and c = 14.442 $Å$ at 294 K as determined by our XRD measurements. The crystal structure is shown in Fig. 1(a). From XRD, we determine a density of $ρ=1.26 g/cm3$⁠, which is close to the literature value of 1.27 g/cm³.²³ In these structures, the basal plane is a (100) facet with the long edge corresponding to the [001] direction.

Brillouin spectra consist of an intense central elastically scattered component (Fig. 2; blocked in these measurements by a shutter) surrounded by anti-Stokes and Stokes doublets, which correspond to the propagating acoustic phonon modes—longitudinal (or quasi-longitudinal, labeled “L”) and transverse (or quasi-transverse, labeled “T, T₁, and T₂”). Each mode is characteristic to the sound velocity in the measured crystal direction. Examples of stacked plots of Brillouin spectra of rubrene as a function of crystal rotation in a 180° backscattering geometry are shown in Figs. 2(a) and 2(b). Because of the symmetry of the photoelastic tensor governing the Brillouin scattering interaction, not all peaks will appear in all spectra. For example, in this orthorhombic crystal, we should expect transverse modes to be nearly invisible at Θ = 0 but perhaps not quite completely invisible due to the sample birefringence and finite convergence and the collection angle of the measurements. Also, in some spectra, the lower-frequency quasi-transverse peak is inseparable from the tails of the elastically scattered peak.

In each measurement, the sample is initially rotated with respect to an angle, Θ. Θ is held fixed, while the sample is rotated in a second direction, α. As the sample is tilted with respect to the incident light, the nonzero Θ value breaks the symmetry, enabling the appearance of the quasi-transverse peaks. At larger $Θ=34°$ (Fig. 2(b)), Brillouin spectra show a shift of the transverse peaks with rotation of the rubrene crystal with respect to the surface normal. This indicates anisotropic elastic properties along the different crystal directions of the basal plane. We measured a total of 35 different combinations of Θ and α, identifying 76 distinct Stokes/anti-Stokes pairs. To expand the range of probed phonon wave vectors, we performed similar measurements on the (001) and (010) surfaces as well (Figs. 2(c) and 2(d), respectively). The rubrene crystal is also tilted at 36° with respect to the c direction, providing both transverse acoustic modes (Fig. 2(e)). Assignments of the transverse and longitudinal peaks are further verified by changing the polarization of the input and output collected light.¹² The sound velocity and elastic stiffnesses are calculated from the Brillouin shifted peaks by solutions of the Christoffel Equations using refractive indices determined from Ref. 24. Full details can be found in the supplementary material. The linear elastic stiffness tensor obtained from these results is (Eq. (1))

The reduced elastic constants, i.e., the plane-stress moduli (Table II), are defined as $c¯αβ=cαβ−cα1*c1β/c11$⁠, where $α,β=2,3$⁠. Polar plots of the sound velocity, V along different planes together with the molecular arrangement, are shown in Fig. 3. From the polar plot in the basal (100) plane, the longitudinal sound velocity reached its maximum at 47° with respect to the b direction.

Using vdw-DFT computational procedures, the calculated elasticity tensor for orthorhombic rubrene is (Eq. (2))

A direct comparison of the computed stiffness tensor (Eq. (2)) with the measured one (Eq. (1)) is not immediately instructive. Instead, a more meaningful analysis may be affected by comparing the effective Young's moduli and reduced (plane-stress) constants listed in Table II. As seen from these data, the vdW-DFT calculations capture the relative trends in Young's moduli correctly although E_x is overestimated by about 50%. The calculated and measured (Brillouin scattering) plane-stress moduli are also in good agreement as are estimates of the anisotropy ratios. Prior data from wrinkling measurements are also generally consistent with this work; the slightly larger anisotropy ratio from wrinkling studies might be a consequence of the difficulty in measuring $c¯33$ accurately due to more pronounced boundary effects that affect wrinkling along the [001] direction (see Fig. 3 in Ref. 10).

An additional and possibly more stringent check on the relative quality of the predictions is the trend in the plane-stress elastic modulus as a function of the in-plane crystal direction. Prior work using the wrinkling technique¹⁰ has shown that the in-plane response of orthorhombic rubrene crystals (within the [010]–[001] habit plane) is distinctly anisotropic with a maximum at approximately $38°$ from the [010] direction. Figure 4 displays this orientation-dependent plane-stress modulus, which is defined as

the angle Θ being measured from the [010] direction, as calculated from Brillouin scattering, wrinkling, and vdW-DFT data. It is indeed noteworthy that all three sets of data are essentially consistent in capturing the orientation-dependent trend in the plane-stress modulus and, moreover, are in fairly close agreement with respect to the maximum stiffness direction. In contrast, the empirical potential used previously¹⁰ only captures stiffness values along the [010] and [001] directions with any degree of accuracy while failing completely in capturing the overall orientation-dependent trend. The need for more accurate descriptions of van der Waals bonding, as provided by the vdW-DFT functionals employed in this work, is clearly borne out for modeling molecular crystals.

In summary, the complete elasticity tensor of rubrene has been determined from Brillouin scattering measurements and vdW-DFT calculations along with Young's moduli and reduced elastic constants. These properties complement previous investigations and may not only prove practical for the study of the strain effect on the carrier mobility²⁵ but also useful for practical applications of rubrene as flexible electronic device components. More generally, the good agreement between the experiment and theory shown here establishes Brillouin scattering and vdW-DFT as potentially valuable characterization techniques for a broad range of crystalline organic semiconductors.

See supplementary material for Brillouin scattering stiffness tensor analysis details.

K.J.K. would like to acknowledge funding from the National Science Foundation Grant DMR-1658019. D.R.M. would like to acknowledge funding from the National Science Foundation Grant CHE-1429086. A.R. acknowledges the research support from the University of Massachusetts Amherst. A.L.B. acknowledges funding from the National Science Foundation Grant DMR-1508627. S.S. acknowledges the research support from Boston University and computational resources at the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.