1. Introduction

2. Method and Computational Procedure

3. Results and Discussion

3.1. Structural Properties of Orthorhombic Rubrene

As shown in Table 1, a lattice constant of 26.79 $\widehat{A}$ (50.8257 a.u.) was obtained and it is in good agreement with the experimental value of Ref. [20]. Table 1 shows the optimized lattice parameters a = 26.7903 Å, b = 7.17001 Å, and c = 14.2112 Å for Orthorhombic Rubrene. These values are in close agreement with those reported in theoretical studies Ref. [33] and Ref. [34].

Table 1. Calculated and measured lattice constant of orthorhombic Rubrene.

Table 1. Calculated and measured lattice constant of orthorhombic Rubrene.

Reference	Lattice Constants			Method/Theory
	a ($\widehat{A}$)	b ($\widehat{A}$)	c ($\widehat{A}$)
Present work	26.79	7.17	14.21	GGA
Ref. [33]	26.86	7.19	14.43	GGA
Ref. [34]	26.660	7.142	14.025	vdW-DFT
Ref. [34]	26.965	7.206	14.442	Experiment at 294K
Ref. [35]	26.789	7.170	14.211	Experiment at 100K
Ref. [36]	14.1289	7.1455	26.7450	GGA

The relaxed structure of orthorhombic Rubrene (Figure 1b) shows the compound to be a polymorph with the Cmca space group. Whenever Rubrene molecules unite to form Orthorhombic crystals, the resulting molecules have a centrosymmetric structure with 2/m symmetry. The cell parameters of Orthorhombic Rubrene crystal (as reported in Table 1) shows the molecules are organized in a herringbone packing pattern with nearly complete $\pi$- $\pi$ stacking in the b direction. This is responsible for its large charge-carrier mobility, which has been affirmed in several pieces of research [16,34]. The mobility of charge carriers in Orthorhombic monocrystals can approach 40 cm²/Vs, which is equivalent to that of amorphous silicon [37,38]. As seen in most organic crystals, Rubrene’s electronic transport is very anisotropic. High mobility values can only be obtained along the lattice’s b axis. The features obtained from Figure 1b agrees reasonably well with the results of Ref. [34] and [39]. The structural properties obtained in this work agree with previous theoretical and experimental values.

3.2. Electronic Properties

Figure 2 depict the density of state and band structure, as well as the state and behaviour of electrons in an Orthorhombic Rubrene crystal. The numerical values for the direct bandgap between the Highest Occupied Molecular Orbit (HOMO) and the Lowest Unoccupied Molecular Orbit (LUMO) are shown in Table 2. The highest valence band results from the highest occupied molecular orbital (HOMO), whereas the lowest conduction band results from the lowest unoccupied molecular orbital (LUMO). Generally, an energetic bandgap of ∼1-5eV has been recorded for OS such as Orthorhombic Rubrene [33], as they tend to have a diminishing density of states around the Fermi energy. This is proportional to the size of the electrical gap, which is fairly considerable. This reduces the density of thermally generated charge carriers in pure organic crystals, such as silicon.

Table 2. Comparison of band gaps (HOMO - LUMO) of orthorhombic Rubrene obtained in this work, experiments and previous calculations.

Table 2. Comparison of band gaps (HOMO - LUMO) of orthorhombic Rubrene obtained in this work, experiments and previous calculations.

Reference	Band Gap		Theory
	Nature	Energy (ev)
Present work	Direct	1.26	PBE
Ref. [36]	Direct	1.13	PBE
Ref. [40]	Direct	1.357	B3LYP 6-311G
Ref. [41]	Direct	2.50	B3LYP/6-311G(d,p)

As shown in the band structure (Figure 2), the bottom of the conduction band (CB) and the top of the valence band (VB) occur at the same momentum value. This implies that the band structure of organic semiconductor Rubrene has a direct bandgap, as can be observed with some other organic semiconductors. The direct bandgap of 1.26 eV obtained within PBE agrees with the calculation of Ref [36] using PBE also with a direct bandgap of 1.13 eV Ref. [41] and [42] found a greater disparity between HOMO and LUMO. Their calculated band gap is 2.50 eV and 2.60 eV respectively at B3LYP/6-311G(d,p) level, which differs from other calculations. So far, there is no theoretical literature report on the consequential variation in the HOMO-LUMO gap [43]. The highest VB’s structural characteristic controls hole transport behaviour. The band splitting of the VB of orthorhombic Rubrene is fairly minimal. Pressure increases intermolecular interaction, resulting in increased mobility [44]. High mobilities are only for holes, and electron mobilities are several orders of magnitude lower [42,45]. Rubrene is a p-type material, as are the vast majority of organic semiconductors [42].

The number of possible electrons (or hole) states per volume at a given energy is given by the density of states as shown in Figure 2. The density of states distribution at the top of VB is relatively smooth in the orthorhombic Rubrene.

The high-symmetry points’ reciprocal coordinates are $\mathrm{\Gamma }$ = (0, 0, 0), X = (0, 0, 0.5), S = (0, 0.5, 0.5), Y = (0, 0.5, 0), $\mathrm{\Gamma }$ = (0, 0, 0), Z = (-0.5, 0, 0), U = (-0.5, 0, 0.5), R = (-0.5, 0.5, 0.5), T = (-0.5, 0.5, 0), and Z = (-0.5, 0, 0). Each band in the band structures occurs in pairs because the orthorhombic Rubrene crystal structure comprises two molecules in a unit cell. Figure 2 depicts the band structure along the k-path $\mathrm{\Gamma }$ – X – S – Y – $\mathrm{\Gamma }$ – Z – U -R – T - Z, where $\mathrm{\Gamma }$ -Z corresponds to the `a’ crystal axis and $\mathrm{\Gamma }$ -Y to the `b’ axis in real space [16].

3.3. Elastic Properties

The values of the computed C_ij show that the Born-Huang Stability criteria have been fully satisfied [46] which proves Rubrene to be mechanically stable. The estimated bulk modulus B, shear modulus G, Young’s modulus E, Poisson ratio, Pugh ratio, and Vickers Hardness using the Voigt-Reuss-Hill approximation are shown in Table 4 [47]. According to [48], a material is considered to be ductile if its bulk to shear modulus ratio, B/G, is more than 1.75; otherwise, it is brittle. The obtained Pugh ratio is 0.74667, which shows Rubrene to be brittle than ductile. This property implies the ability to break with minimal elastic deformation when stressed and no considerable plastic deformation. Brittle materials have more strength than ductile materials. Brittle materials are more resistant to compression. Even high-strength brittle materials absorb relatively little energy before breakage. This feature proves the ability of orthorhombic Rubrene as an organic semiconductor useful in redefining the future of flexible and stretchable electronics. Furthermore, the Poisson’s ratio can also prove the ductility of a material if the ratio is greater than 0.26 and brittleness is less than 0.26. The obtained Poisson ratio is 0.03292, which agrees with the Pugh ratio can classify orthorhombic Rubrene as brittle.

Table 3. Comparison of independent elastic constants of orthorhombic Rubrene obtained in this work, experiment and previous calculations.

Table 3. Comparison of independent elastic constants of orthorhombic Rubrene obtained in this work, experiment and previous calculations.

Reference	This work	Ref. [20]	Ref. [34]	Ref. [34]
Method	PBE	AIREBO	vdw-DFT	Experiment
C11 (GPa)	18.8	15.54	25.31	18.48
C12 (GPa)	-8.7	1.08	6.94	2.63
C13 (GPa)	1.6	2.08	6.78	7.68
C22 (GPa)	13.6	17.85	16.99	13.39
C23 (GPa)	9.4	10.82	10.53	7.77
C33 (GPa)	14.6	13.29	13.94	14.32
C44 (GPa)	7.2	2.03	6.66	6.46
C55 (GPa)	13.2	1.97	4.41	2.8
C66 (GPa)	6.5	3.36	3.67	6.8

The Vickers hardness H_v is obtained as shown in Table 4. The hardness of a material measures its resistance to plastic deformation produced by applied forces. The result helps characterise the elastic and plastic properties of a solid. The predicted Vickers hardness H_v of 1.080 GPa was obtained for orthorhombic Rubrene using Chen’s model [49]. The result shows that orthorhombic Rubrene is far from hard and cannot be classified as superhard.

Elastic Constants $C_{ij}$ (GPa) = $\left[\begin{array}{cccccc}18.846944& -8.738236& 1.566589& 0.00000& 0.00000& 0.00000\\ -8.738236& 13.633697& 9.433130& 0.00000& 0.00000& 0.00000\\ 1.566589& 9.433130& 14.599899& 0.00000& 0.00000& 0.00000\\ 0.00000& 0.00000& 0.00000& 7.220388& 0.00000& 0.00000\\ 0.00000& 0.00000& 0.00000& 0.00000& 13.187243& 0.00000\\ 0.00000& 0.00000& 0.00000& 0.00000& 0.00000& 6.507988\end{array}\right]$ Calculated elasticity tensor for orthorhombic Rubrene

Also, the calculated reduced elastic constants and anisotropy ratio (C₂₂/C₃₃) of orthorhombic Rubrene obtained in this work is shown in Table 5. The values of reduced constants C₂₂, C₃₃, and the anisotropy ratio (C₂₂/C₃₃) are in close agreement with the result of Ref [34] from the experimental and vdw-DFT method. These findings may be beneficial not only for studying the strain effect on carrier mobility [16], but also for Rubrene’s actual use as a flexible electrical device [34].

Its directional elastic properties need to be analysed and visualised to better understand an anisotropic material such as Rubrene. This includes the young modulus, linear compressibility, shear modulus and Poisson ratio as shown in Figure 5. The debye temperature and average Debye sound velocity recorded in this research for orthorhombic Rubrene are 331.008K and 2384.484m/s, respectively (Table 4).

Table 4. Calculated Voigt-Reuss-Hill approximation moduli for orthorhombic Rubrene in this work

Table 4. Calculated Voigt-Reuss-Hill approximation moduli for orthorhombic Rubrene in this work

Method	PBE
Bulk Modulus (GPa)	4.163
Shear Modulus (GPa)	11.519
Young Modulus (GPa)	5.576
Poisson Ratio	0.03292
Pugh Ratio (B/G)	0.74667
Vickers Hardness (GPa)	1.08
Average Debye sound velocity ( m/s )	2384.484
Debye temperature (K)	331.008

Table 5. Calculated reduced elastic constants and anisotropy ratio (C₂₂/C₃₃) of orthorhombic Rubrene obtained in this work, experiments and previous calculations.

Table 5. Calculated reduced elastic constants and anisotropy ratio (C₂₂/C₃₃) of orthorhombic Rubrene obtained in this work, experiments and previous calculations.

Reference	Method	C22(GPa)	C33 (GPa)	Anisotropy ratio (C22/C33)
This work	PBE	13.6	14.6	0.93
[34]	vdw-DFT	15.08	12.12	1.24
[34]	Experiment	13.02	11.13	1.17

Young modulus is a mechanical property that measures the tensile/rigidity or stiffness of a material when the force is applied. It implies the ratio of the tensile stress to the proportional deformation/tensile strain.

$$\begin{array}{c}\hfill E=\frac{\sigma }{\epsilon }\end{array}$$

(1)

Where tensile stress $\sigma =\frac{F}{A}$ is the force per unit area and tensile strain $\epsilon =\frac{dl}{l}$ is extension per unit length. The young modulus reported for orthorhombic Rubrene in this work is 55.76 kbar which shows Rubrene as a non-rigid material (Figure 4a).

Linear compressibility or the bulk modulus helps describe a material’s behaviour when pressure is applied, which can be either negative or positive. The 2D and 3D surface plot (Figure 4b and Figure 5b) shows the positive values of linear compressibility plotted in green and the negative value in red. The result indicates that orthorhombic Rubrene crystal structure exhibits negative linear compressibility.

Figure 4. (a)Young’s Modulus, (b)Linear Compressibility, (c)Shear Modulus, and (d)Poisson Ratio’s in 2D showing its directional planes

Figure 4. (a)Young’s Modulus, (b)Linear Compressibility, (c)Shear Modulus, and (d)Poisson Ratio’s in 2D showing its directional planes

Figure 5. (a)Young’s Modulus, (b)Linear Compressibility, (c)Shear Modulus, and (d)Poisson Ratio’s in 3D showing its directional planes

Figure 5. (a)Young’s Modulus, (b)Linear Compressibility, (c)Shear Modulus, and (d)Poisson Ratio’s in 3D showing its directional planes

Shear modulus is the ratio of shear stress to the shear strain. It gives information on how resistant a material is to deformations as shown in Figure 4c and Figure 5c. The 2D and 3D plots of the Poisson ratio Figure 4d and Figure 5d showed the lateral strain and the longitudinal strain on orthorhombic Rubrene. Poisson ratio for elastic materials is the ratio of the lateral strain and longitudinal strain, which gives information on how materials deform under loading. However, the calculated Poisson ratio value is 0.03292, which lies between -1 to 0.5. The result showed the pristine form of orthorhombic Rubrene as almost perfectly incompressible as there is little or no transverse deformation when axial strain is applied.

3.4. Temperature and Doping Dependent Properties

To address the thermoelectric properties, the electronic fitness test is crucial as it evaluates the thermopower for arbitrary band structures and conductivity [32]. Suitable thermoelectric materials, in general, have intricate electronic structures that aren’t described by a simple parabolic band [32]. Therefore, electronic fitness function calculation is necessary to identify such materials and address the $\sigma$ (Conductivity) and S (Seebeck coefficient) conflict.

The Electronic Fitness Function (EFF) displayed in Figure 6 shows an increase with temperature for both concentrations, which is caused by the rising temperature and which improves its thermoelectric performance. There is a dip at carrier concentration 6.5 x 10²¹ for the hole concentration and reaches peak at 2 x 10²¹ and 1.1 x 10²². This work also predicted that orthorhombic Rubrene has a high electronic fitness function at about 500K. Organic semiconductors like Rubrene are mostly p-type semiconductors.

The structure for the EFF in Figure 6 shows larger p-type EFF values at high doping levels. This is attributed to the secondary conduction band contribution in Rubrene. Furthermore, as the temperature rises, it exhibits improved p-type performance. Valley anisotropy, higher band degeneracy, and multiband contributions in valence bands at higher energies, particularly at high doping levels, result in larger EFF in p-type materials. The features observed from these results show orthorhombic rubrene as a promising organic material for thermoelectric applications.

Figure 7 depicts the transport inverse effective mass as a function of carrier concentrations ranging from 300K to 800K. The result shows it exhibits a light, effective mass of holes. This agrees with the experiments performed in Ref [50].

Figure 7. Inverse Effective Mass as a function of the Carrier Concentrations from 300K to 800K

Figure 7. Inverse Effective Mass as a function of the Carrier Concentrations from 300K to 800K

Figure 8. Transport properties of Rubrene showing the power factor, electrical conductivity and Seebeck coefficient, respectively.

Figure 8. Transport properties of Rubrene showing the power factor, electrical conductivity and Seebeck coefficient, respectively.

4. Conclusion

References

1. Jung, J.; Ulański, J. , Charge Carrier Transport in Organic Semiconductor Composites – Models and Experimental Techniques. In Solution-Processable Components for Organic Electronic Devices; John Wiley and Sons, Ltd, 2019; chapter 6, pp. 309–363, [https://onlinelibrary.wiley.com/doi/pdf/10.1002/9783527813872.ch6]. [CrossRef]
2. Ahmad, S. Organic semiconductors for device applications: current trends and future prospects. Journal of Polymer Engineering 2014, 34, 279–338. [Google Scholar] [CrossRef]
3. Dey, A.; Singh, A.; Das, D.; Iyer, P. Organic Semiconductors: A New Future of Nanodevices and Applications 2015. pp. 97–128. [CrossRef]
4. Diemer, P.J.; Harper, A.F.; Niazi, M.R.; Petty II, A.J.; Anthony, J.E.; Amassian, A.; Jurchescu, O.D. Laser-Printed Organic Thin-Film Transistors. Advanced Materials Technologies 2017, 2, 1700167. [Google Scholar] [CrossRef]
5. Kim, J.T.; Lee, J.; Jang, S.; Yu, Z.; Park, J.; Jung, E.; Lee, S.; Song, M.H.; Whang, D.R.; Wu, S.; Park, S.H.; Chang, D.; Lee, B.R. Solution Processable Small Molecules as Efficient Electron Transport Layers in Organic Optoelectronic Devices. Journal of Materials Chemistry A 2020, 8. [Google Scholar] [CrossRef]
6. Riede, M.; Lüssem, B.; Leo, K. Organic Semiconductors. Comprehensive Semiconductor Science and Technology, p. 448–507. [CrossRef]
7. Walker, A.B. Multiscale Modeling of Charge and Energy Transport in Organic Light-Emitting Diodes and Photovoltaics. Proceedings of the IEEE 2009, 97, 1587–1596. [Google Scholar] [CrossRef]
8. Chen, F.C. Organic Semiconductors. Encyclopedia of Modern Optics, p. 220–231. [CrossRef]
9. El-Saba, M. , Carrier Transport in Organic Semiconductors and Insulators; 2017. [CrossRef]
10. Kim, J.; Yasuda, T.; Yang, Y.; Adachi, C. Advanced Materials 2013, 25, 2666–2671. [CrossRef]
11. Choi, M.; Lee, H.N. Light-Emission and Electricity-Generation Properties of Photovoltaic Organic Light-Emitting Diodes with Rubrene/DBP Light-Emission and Electron-Donating Layers. International Journal of Photoenergy 2014, 2014, 361861. [Google Scholar] [CrossRef]
12. Saxena, K.; Mehta, D.; Rai, V.K.; Srivastava, R.; Chauhan, G.; Kamalasanan, M.; Jain, V. Studies on organic light-emitting diodes based on rubrene-doped zinc quinolate. Physica Status Solidi (a) 2009, 206, 1660–1663. [Google Scholar] [CrossRef]
13. Weinberg-Wolf, J.R.; McNeil, L.E.; Liu, S.; Kloc, C. Evidence of low intermolecular coupling in rubrene single crystals by Raman scattering. Journal of Physics: Condensed Matter 2007, 19, 276204. [Google Scholar] [CrossRef]
14. Sai, N.; Tiago, M.; Chelikowsky, J.; Reboredo, F. Optical spectra and exchange-correlation effects in molecular crystals. Physical Review B, 77. [CrossRef]
15. Wikipedia contributors. Rubrene — Wikipedia, The Free Encyclopedia, 2021. [Online; accessed 11-February-2022].
16. Reyes-Martinez, M.; Crosby, A.; Briseno, A. Nat. Commun, 6, 6948.
17. Lin, K.; Wang, Y.; Chen, K.; Ho, C.; Yang, C.; Shen, J.; Chiu, K. Scientific Reports 2017, 7. Funding Information: We gratefully acknowledge financial supports from Chung Yuan Christian University (Grant number CYCU-107011022) and from the Ministry of Science and Technology of ROC (Grant numbers MOST 104-2112-M-033-005 and MOST 105-2112-M-033-008). Publisher Copyright: © The Author(s) 2017. [CrossRef]
18. Majewska, N.; Gazda, M.; Jendrzejewski, R.; Majumdar, S.; Sawczak, M.; Śliwiński, G. Organic semiconductor rubrene thin films deposited by pulsed laser evaporation of solidified solutions. Proc. SPIE 10453, Third International Conference on Applications of Optics and Photonics, Vol. 104532H. [CrossRef]
19. Zeng, X.; Zhang, D.; Duan, L.; Wang, L.; Dong, G.; Qiu, Y. Morphology and fluorescence spectra of rubrene single crystals grown by physical vapor transport. Applied Surface Science, 253, 6047–6051. [CrossRef]
20. Reyes-Martinez, M.; Ramasubramaniam, A.; Briseno, A.; Crosby, A. The Intrinsic Mechanical Properties of Rubrene Single Crystals. Advanced Materials, 24, 5548–5552. [CrossRef]
21. Monkhorst, H.J.; Pack, J.D. Special points for Brillouin-zone integrations. Phys. Rev. B 1976, 13, 5188–5192. [Google Scholar] [CrossRef]
22. Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed]
23. Perdew, J.P.; Chevary, J.A.; Vosko, S.H.; Jackson, K.A.; Pederson, M.R.; Singh, D.J.; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 1992, 46, 6671–6687. [Google Scholar] [CrossRef] [PubMed]
24. Perdew, J.P.; Ruzsinszky, A.; Csonka, G.I.; Vydrov, O.A.; Scuseria, G.E.; Constantin, L.A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406. [Google Scholar] [CrossRef] [PubMed]
25. Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864–B871. [Google Scholar] [CrossRef]
26. Kohn, W.; Sham, L.J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133–A1138. [Google Scholar] [CrossRef]
27. Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G.; Cococcioni, M.; Dabo, I. Quantum Espresso: a modular and open-source software project for quantum simulations of materials. Journal of physics: Condensed matter, 21, 395502.
28. Giannozzi, P.; Andreussi, O.; Brumme, T.; Bunau, O.; Nardelli, M.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Cococcioni, M. Advanced capabilities for materials modelling with Quantum ESPRESSO. Journal of Physics: Condensed Matter, 29, 465901.
29. Vanderbilt, D. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Phys. Rev. B 1990, 41, 7892–7895. [Google Scholar] [CrossRef] [PubMed]
30. Baroni, S.; De Gironcoli, S.; Dal Corso, A.; Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Reviews of Modern Physics, 73, 515.
31. Louail, L.; Maouche, D.; Roumili, A.; Sahraoui, F. Calculation of elastic constants of 4d transition metals. Mater. Lett, 58, 2975–2978.
32. Xing, G.; Sun, J.; Li, Y.; Fan, X.; Zheng, W.; Singh, D. The thermoelectric fitness function. Phys. Rev. Mater, 1, 065405, 079901.
33. Wang, D.; Tang, L.; Long, M.; Shuai, Z. First-principles investigation of organic semiconductors for thermoelectric applications. The Journal of Chemical Physics, 131, 224704. [CrossRef]
34. Zhang, Y.; Manke, D.; Sharifzadeh, S.; Briseno, A.; Ramasubramaniam, A.; Koski, K. The elastic constants of rubrene determined by Brillouin scattering and density functional theory. Applied Physics Letters, 110, 071903. [CrossRef]
35. Jurchescu, O.; Meetsma, A.; Palstra, T.; Crystallogr, A. Sect. B: Struct. Sci, 62, 330.
36. Yanagisawa, S.; Morikawa, Y.; Schindlmayr, A. HOMO band dispersion of crystalline rubrene: Effects of self-energy corrections within the GW approximation. Physical Review B, 88. [CrossRef]
37. Podzorov, V.; Menard, E.; Borissov, A.; Kiryukhin, V.; Rogers, J.; Gershenson, M. Intrinsic charge transport on the surface of organic semiconductors. Phys. Rev. Lett, 93.
38. Yamagishi, M.; Takeya, J.; Tominari, Y.; Nakazawa, Y.; Kuroda, T.; Ikehata, S.; Uno, M.; Nishikawa, T.; Kawase, T. High-mobility double-gate organic single-crystal transistors with organic crystal gate insulators. Appl. Phys. Lett, 90, 182117.
39. Fumagalli, E.M. Growth and physical properties of crystalline rubrene. Università degli Studi di Milano-Bicocca. Doctorate in Materials Science.
40. Musa, A.; Gidado, A.; Mohammed, L.; Yunusa, K.; Suleiman, A. Molecular and Electronic Properties of Rubrene and Its Cyanide Derivative Using Density Functional Theory (DFT.
41. Zhang, M.; Hua, Z.; Liu, W.; Liu, H.; He, S.; Zhu, C.; Zhu, Y. A DFT study on the photoelectric properties of rubrene and its derivatives. Journal of Molecular Modeling, 21. [CrossRef]
42. Missaoui, A.; Khabthani, J.; Laissardière, G.; Mayou, D. Two-Dimensional Electronic Transport in Rubrene: The Impact of Inter-Chain Coupling. Entropy, 26. [CrossRef]
43. Mukherjee, T.; Sinha, S.; Mukherjee, M. Electronic structure of twisted and planar rubrene molecules: a density functional study. Physical Chemistry Chemical Physics, 20, 18623–18629. [CrossRef]
44. Rang, Z.; Nathan, M.; Ruden, P.; Podzorov, V. ; Gershenson.
45. Bisri, S.; Takenobu, T.; Takahashi, T.; Iwasa, Y. Electron transport in rubrene single-crystal transistors. Appl. Phys. Lett, 96, 183304.
46. Fan, Q.; Wei, Q.; Yan, H.; Zhang, M.; Zhang, D.; Zhang, J. A New Potential Superhard Phase of OsN 2. Acta Physica Polonica, A, 126.
47. Hill, R. Proceedings of the Physical Society. Section A, A65, 349. [CrossRef]
48. Pugh, S. XCII. Relations between the Elastic Moduli and the Plastic Properties of Polycrystalline Pure Metals. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 45, 823–843.
49. Modeling hardness of polycrystalline materials and bulk metallic glasses. Intermetallics 2011, 19, 1275–1281. [CrossRef]
50. Machida, S.; Nakayama, Y.; Duhm, S.; Xin, Q.; Funakoshi, A.; Ogawa, N.; Ishii, H. Highest-Occupied-Molecular-Orbital Band Dispersion of Rubrene Single Crystals as Observed by Angle-Resolved Ultraviolet Photoelectron Spectroscopy. Physical Review Letters, 104. [CrossRef]