A first-principles study of the vibrational properties of crystalline tetracene under pressure

TL;DR: The results suggest that the experimentally reported improvement of the photocurrent under pressure may be ascribed to an increase in intermolecular interactions as also the dielectric tensor.

Abstract: We present a comprehensive study of the hydrostatic pressure dependence of the vibrational properties of tetracene using periodic density-functional theory (DFT) within the local density approximation (LDA). Despite the lack of van der Waals dispersion forces in LDA we find good agreement with experiment and are able to assess the suitability of this approach for simulating conjugated organic molecular crystals. Starting from the reported x-ray structure at ambient pressure and low temperature, optimized structures at ambient pressure and under 280 MPa hydrostatic pressure were obtained and the vibrational properties calculated by the linear response method. We report the complete phonon dispersion relation for tetracene crystal and the Raman and infrared spectra at the centre of the Brillouin zone. The intermolecular modes with low frequencies exhibit high sensitivity to pressure and we report mode-specific Gruneisen parameters as well as an overall Gruneisen parameter [Formula: see text]. Our results suggest that the experimentally reported improvement of the photocurrent under pressure may be ascribed to an increase in intermolecular interactions as also the dielectric tensor.

Summary

1. Introduction

• Organic semiconductors have great potential as active materials in optoelectronic devices.
• A recent DFT study performed on tetracene and other molecular crystals, and containing a correction for the vdW interactions, correctly predicts the structural, electronic and optical properties of the crystals [19, 20].
• So for this study the authors return to the local density approximation (LDA).
• Though it lacks a physically correct description of vdW dispersion forces, this is compensated by a tendency to overbind which usually gives fairly good predictions of equilibrium geometry in weakly-bonded molecular crystals.
• However this has not yet been examined in detail.

2. Method and Computational Details

• The authors calculations used the CASTEP code [27] which implements the plane-wavepseudopotential formulation of DFT together with density-functional perturbation theory (DFPT).
• The two structures converged to the same minimum total energy within 6 meV per unit cell and to equivalent unit cell parameters, in agreement with reference [33].
• Calculations were performed on a unit cell of this tetracene crystal containing two symmetrically inequivalent molecules so that the unit cell contains a total of 36 carbon and 24 hydrogen atoms.
• The coarse grid contains 14 irreducible q-points.

3.1 Optimisation of Tetracene Structure at Ambient and 280 MPa Pressures

• Table 1 presents DFT-LDA results for unconstrained optimisation of the LT structure at ambient pressure conditions together with the experimentally reported LT data [31].
• In accordance with the well-known tendency of the LDA to overbind the atoms [40, 41], the volume of the DFT-LDA optimised structure at ambient pressure was smaller by ~11% than the experimentally measured volume, despite the lack of vdW forces.
• Upon optimisation, the lengths of all the lattice vectors decreased, with a maximum reduction in b of ~5%, but the cell angles did not change appreciably.
• The overlap between the molecular wavefunctions in the ab plane increases.
• Upon the application of a small amount of pressure, the interlayer distance Z is almost unchanged, but there is a slight increase in the angle between the planes of the adjacent molecules from 150.1° to 150.6°.

3.2. Phonon Dispersion Curves

• Figure 2 displays the phonon dispersion curves 𝜔(𝑞) computed for the two optimised structures of tetracene along a path that includes the high symmetry q-points and a few others in the triclinic unit cell.
• The intermolecular and low-lying intramolecular modes shown in Figure 2 (a,b) are highly dispersive, while the higherfrequency intramolecular modes shown in Figure 2 (c,d) exhibit much less dispersion.
• Figure 4 shows schematic illustrations of the atomic and molecular displacement associated with some of the DFT-LDA calculated phonons below 300 cm -1 .
• Further detailed information on the displacement of atoms corresponding to each of the 180 modes is available in the supplementary information.
• In general, modes are shifted to higher frequencies when tetracene is compressed, indicating that the crystal becomes stiffer.

3.3 Grüneisen Parameters

• Grüneisen parameters for each phonon mode are defined by the dependence of the phonon frequencies on the crystal volume.
• The overall crystal Grüneisen parameter is evaluated by averaging the individual mode-Grüneisen parameters, weighted by their specific heat contributions (see supplementary information).
• The overall crystal Grüneisen parameter of tetracene computed by DFT-LDA is 2.82, which is ‘qualitatively’ similar to that measured and calculated as 3.6 and 3.46, respectively for naphthalene [50].
• Figure 6(b) shows the modeGrüneisen parameters at the zone-centre for the modes below 600 cm -1 .
• It may be seen that modes with frequency below 155 cm -1 are the most pressure sensitive, with Grüneisen parameters around 3, decreasing to less than 1 at higher frequencies.

3.4 Pressure Dependence of Raman and Infrared Spectra

• The modes of the DFT-LDA-calculated phonons were assigned using symmetry group analysis for tetracene, , which predicts a total of 90 modes for each of the symmetry modes Au and Ag respectively.
• The frequencies of the low-lying Raman-active modes at ambient pressure conditions calculated using DFT-LDA plotted against the experimentally measured frequencies in Ref.[25] show excellent agreement, as indicated by the slope of (1.04±0.02) and correlation coefficient of 0.998 (see supplementary information).
• A small scaling error – in this case 1.04 is typical and expected of the comparison between DFT and experimental frequencies.
• This upshift is non-uniform, 1P 1iC showing some variance among the mode Grüneisen parameters.
• This work additionally predicts several bands in the intermolecular low frequency region below 160 cm -1 .

3.5 Pressure Dependence of Dielectric Constant

• The propagation of radiation through the crystal subject to a low-frequency (IR and lower) external electric field is described by the static dielectric tensor (𝜀0) whilst the optical tensor (𝜀∞) describes the interaction of the crystal with a high-frequency electric field.
• Because of the triclinic symmetry of the tetracene crystal, both the static and optical dielectric tensors are anisotropic with three distinct diagonal components 𝜀𝑥𝑥 , 𝜀𝑦𝑦 and 𝜀𝑧𝑧 .
• The DFT-LDA calculated optical dielectric constant at ambient pressure is higher than the experimentally reported values due to the underestimation of the band gap, which is inversely proportional to 𝜀(𝜔) [57, 58].
• This shows that 𝜀(𝜔) is sensitive to the change that occurs along the b direction within the ab herringbone plane in the unit cell rather than along the c direction of the stacking layers.
• Furthermore, the increase in the transfer integral leads to an increase in the mobility of charge carriers in accord with the experimental observation [21].

4. Conclusions

• The applicability of DFT-LDA to probe the pressure dependence of structural and vibrational properties of organic molecular crystals has been assessed through the calculation of tetracene properties.
• In spite of the lack of account of vdW dispersion forces and the tendency of DFT-LDA to overbind atoms, resulting in smaller calculated volumes than found in experiment, there is good agreement for the vibrational properties and associated Raman and infrared spectra between theory and experiment.
• For the first time, the complete set of phonon dispersion relations have been presented for tetracene enabling computation of the Grüneisen parameter.
• As indicated by the mode-specific Grüneisen parameters, the intermolecular modes were more sensitive to the applied pressure than the intramolecular modes as expected.
• The DFT-LDA results also reveal that application of hydrostatic pressure increases the dielectric tensors anisotropically with a maximum change exhibited along the b axis.

Figures

Figure 6. (a) The average mode-specific Grüneisen parameter of tetracene crystal as a function of mode frequency for the intermolecular low frequency modes. (b) The modespecific Grüneisen parameter of tetracene crystal at as a function of mode frequency for the modes with frequencies below 600 cm -1 .

Figure 7. The Raman spectra of LT tetracene structure at ambient pressure conditions (solid line) and under hydrostatic pressure of 280MPa (dotted line).

Figure 2. Theoretically DFT-LDA calculated dispersion relations for LT tetracene along the principal symmetry directions and some other points in BZ along the path described in Figure 2. The intermolecular modes in (a) and (b) are at ambient pressure and under hydrostatic pressure of 280MPa, respectively. The intermolecular and low-laying intramolecular modes in (c) and (d) are at ambient pressure and under hydrostatic pressure of 280MPa, respectively. The high symmetry points of the BZ are labelled relative to the cell vectors according to reference [37].

Table 1. The DFT-LDA lattice parameters for tetracene computed at ambient and 280 MPa hydrostatic pressures together with those experimentally measured in Refs. [24] and [31]. The area of the ab plane is defined as (Aab =ab sin).

Table 2. Comparison of the frequencies of the intermolecular and some of the intramolecular infrared modes (in cm -1 ) calculated using DFT-LDA (this work) and reported experimental and theoretically calculated data from literature [52-56]. R denotes predicted Raman-active band.For complete infrared DFT-LDA calculated modes see supplementary information.

Figure 1. The structure of tetracene crystal at ambient pressure and a temperature of 175K. The image illustrates the stacking of the layers along the c-axis.

Figure 8. The infrared spectra of LT tetracene structure at ambient pressure conditions and under hydrostatic pressure of 280MPa.

Figure 5. Theoretically DFT-LDA calculated vibrational density of states for LT tetracene at ambient pressure and under hydrostatic pressure of 280MPa.

Table 3: The complete dielectric tensor together with the eigenvalues for the optical frequencies () and the low frequencies () for DFT-LDA optimised LT tetracene structure at ambient and 280MPa pressures. Also the dielectric Grüneisen parameter is shown. The Cartesian x-axis is taken to be along the crystal x-direction parallel to the lattice vector a.

Full text

A First Principles Study of the Vibrational Properties of Crystalline
Tetracene under Pressure
Mayami Abdulla
1
, Keith Refson
2,3
, Richard H. Friend
1
, Peter D. Haynes
4
1
Department of Physics, University of Cambridge, Cavendish Laboratory, 19 J. J. Thomson
Avenue, Cambridge CB3 0HE, UK
2
ISIS Facility, STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11
0QX, UK
3
Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX
4
Departments of Materials and Physics, Imperial College London, Exhibition Road, London
SW7 2AZ, UK
Abstract
We present a comprehensive study of the hydrostatic pressure dependence of the vibrational
properties of tetracene using periodic density-functional theory (DFT) within the local
density approximation (LDA). Despite the lack of van der Waals dispersion forces in LDA
we find good agreement with experiment and are able to assess the suitability of this
approach for simulating conjugated organic molecular crystals. Starting from the reported X-
ray structure at ambient pressure and low temperature, optimised structures at ambient
pressure and under 280 MPa hydrostatic pressure were obtained and the vibrational properties
calculated by the linear response method. We report the complete phonon dispersion relation
for tetracene crystal and the Raman and infrared spectra at the centre of the Brillouin zone.
The intermolecular modes with low frequencies exhibit high sensitivity to pressure and we
report mode-specific Grüneisen parameters as well as an overall Grüneisen parameter γ=2.8.
Our results suggest that the experimentally reported improvement of the photocurrent under
pressure may be ascribed to an increase in intermolecular interactions as also the dielectric
tensor.

2
1. Introduction
Organic semiconductors have great potential as active materials in optoelectronic devices.
Among these materials, molecular semiconductors of the oligoacene family have attracted
attention due to their promising high charge carrier mobilities, which enable them to be
incorporated in functional devices such as light-emitting diodes [1, 2], field effect transistors
[3-8] and photovoltaics [9-12]. The molecules within oligoacene crystals are bound together
by van der Waals (vdW) dispersion forces. Therefore the crystal is compliant, so that the
application of a small external pressure can significantly alter structural and electronic
properties whilst preserving the chemical structure of the molecule. The interest in molecular
crystals along with the pressure dependence of their properties dictates the need to correctly
predict the properties of the crystal under ambient pressure conditions and when subjected to
external pressure.
Density-functional theory (DFT) is by far the most popular method for performing first-
principles quantum-mechanical simulations of materials, because it balances a sufficiently
accurate treatment of exchange and correlation for many purposes with a moderate
computational cost. However the usual approximations employed to describe exchange and
correlation – the generalised gradient approximations (GGAs) – lack any account of vdW
dispersion forces that play an important role in carbon-based conjugated semiconductors.
Semi-empirical [13-16] and first-principles [17] descriptions of vdW interactions are under
development, and the reader is advised to review Klimeš et al. for further information [18]. A
recent DFT study performed on tetracene and other molecular crystals, and containing a
correction for the vdW interactions, correctly predicts the structural, electronic and optical
properties of the crystals [19, 20]. However extensions of these methods to phonon properties
are not yet widely available, so for this study we return to the local density approximation

3
(LDA). Though it lacks a physically correct description of vdW dispersion forces, this is
compensated by a tendency to overbind which usually gives fairly good predictions of
equilibrium geometry in weakly-bonded molecular crystals. It might be anticipated that
vibrational properties including intermolecular phonon modes may also benefit from this
error cancellation, as they are defined by small perturbations around the equilibrium
geometry. However this has not yet been examined in detail. It is important to emphasise that
both the size of the LDA binding energy and the asymptotic behaviour are incorrect and thus
its accuracy for thermochemical properties is inconsistent.
Among oligoacene crystals, tetracene (C
18
H
12
), also known as naphthacene or 2,3-
benzanthracene, is a good choice for such a study due to the availability in the literature of
experimental results under ambient and high pressures. Tetracene is a polycyclic aromatic
hydrocarbon that consists of four fused benzene rings in a planar structure. The low
temperature unit cell of polymorph-I of the crystal contains two molecules, which are
arranged in herringbone layers in the ab plane and stacked along the c-axis as shown in
Figure 1. The transport of charge carriers in tetracene depends upon both the internal
molecular structure and the molecular packing within the crystalline state. A previous study
demonstrates an increase of charge carrier mobility with pressure in tetracene, pentacene and
the tetracene derivative rubrene, within the same polymorph [21, 22]. Tetracene also exhibits
a sharp decrease in resistivity as the pressure increases up to 20 GPa [23]. A detailed
experimental and theoretical study of the structural and vibrational properties under
hydrostatic pressure revealed anisotropic changes in the lattice constants and vibrational
modes [24]. Furthermore, the Raman phonon frequencies as a function of temperature and
pressure were measured for the two polymorphs and the results were matched with
theoretically calculated frequencies for isolated tetracene molecules [25]. All DFT studies of
tetracene reported in the literature were performed using a single isolated molecule rather

4
than on a periodic crystal, thereby excluding the low frequency intermolecular vibrational
modes that are sensitive to the change in volume and intermolecular interactions [25, 26].
This paper presents a detailed first-principles study of the tetracene molecular crystal under
hydrostatic pressure. To our knowledge this is the first study of vibrational properties of an
oligoacene that has been conducted on a crystal rather than a molecule. Commencing from
experimentally reported X-ray structures of tetracene crystal, the properties of tetracene were
calculated under ambient and 280 MPa hydrostatic pressures.
The paper is arranged as follows: Section 2 describes the computational methodology.
Section 3 reports the DFT-LDA results for the structure (Sec. 3.1), phonon dispersion (Sec.
3.2), pressure dependence of the Grüneisen parameter (Sec. 3.3), Raman and infrared spectra
(Sec. 3.4), and the pressure dependence of the dielectric constant (Sec. 3.5). Whenever
possible, the DFT-LDA results are compared with corresponding experimental and
theoretical data available in the literature. Section 4 presents the conclusions.

5
2. Method and Computational Details
Our calculations used the CASTEP code [27] which implements the plane-wave-
pseudopotential formulation of DFT together with density-functional perturbation theory
(DFPT). Exchange and correlation were treated within the LDA [28, 29]. Initially, the two
available X-ray crystal structures of tetracene labelled TETCEN [30], recorded at room
temperature (RT), and TETCEN01 [31], recorded at a low temperature (LT) of 175 K,
available from the Cambridge Structural Database (CSD) [32] were geometry-optimised. The
two structures converged to the same minimum total energy within 6 meV per unit cell and
to equivalent unit cell parameters, in agreement with reference [33]. However, the DFT-LDA
phonon dispersion calculation for the RT structure resulted in imaginary frequencies for some
acoustic modes, indicating the instability of this structure at 0 K. It is likely that vibrational
entropy – which is not taken into account in the DFT optimisation – may stabilise this phase
at RT. Accordingly, the calculations presented here proceeded using the LT structure. Figure
1 displays the crystal structure as experimentally determined at LT.
Calculations were performed on a unit cell of this tetracene crystal containing two
symmetrically inequivalent molecules so that the unit cell contains a total of 36 carbon and
24 hydrogen atoms. Optimised norm-conserving pseudopotentials were used to describe the
atomic nuclei and core electrons [34], and the electronic wavefunction was expanded using a
plane-wave basis with cut-off energy of 780 eV. The convergence tolerance for electronic
total energy minimisation was 10
-10
eV/atom and the k-point sampling of the Brillouin zone
(BZ) used a 2×2×1 mesh according to the Monkhorst-Pack (MP) scheme [35]. Unconstrained
variable-cell geometry optimisations of the tetracene crystal unit cell were performed a using
the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) [36]. As is typical for weakly
bonded molecular crystals, stringent geometry optimisation convergence criteria were