Abstract
1. Introduction
The success of graphene have triggered the tide of searching novel two-dimensional (2D) materials with desired properties[
Recently, a new 2D carbon allotrope named C568 has been predicted, which is composed of 5, 6, and 8-membered rings of carbon[
2. Computational methods
The electronic structures of C568 are calculated through density functional theory, which is realized in the Vienna abinitio Simulation Package (VASP)[
3. Results and discussion
3.1. The crystalline structure and electronic structures of C568
To study the electronic structures of C568, the crystalline structure of C568 has been optimized (Fig. 1(a)).
Figure 1.(Color online) (a) The crystalline structures and (b) 2D charge density difference plot of C568. The square unit-cell marked in green line, where
As seen, C568 is formed of 5-, 6-, and 8-membered rings, which are composed of sp2- and sp3-hybridized carbon atoms. Different from sp2 hybridized graphene, the carbon rings in C568 are not in the same plane, due to the presence of sp3-hybridization. C568 possesses a space group of P-4m2(115) with tetragonal lattice. The optimized C568 unit cell contains 13 carbon atoms with lattice parameters of a = b = 5.725 Å, which is consistent with the values reported in a recent work[
The electronic band structure is presented in Fig. 2. The C568 possesses an indirect band structure with a band gap of 1.06 eV, which is consistent with the recent work[
Figure 2.(Color online) The band structures and projected density of states of C568, with the Fermi level are set to zero. The charge densities of several key states are also presented.
3.2. The external strain effects
External strain is one of the most common effects in the practical applications of materials, which can significantly modify the electronic properties of materials. To understand possible effects of the external strain effect on the electronic properties of C568, we have considered in-plane uniaxial strain and biaxial strain. Through calculations, three unique properties have been found: 1) an indirect-to-direct band gap transition occurs under compressive strain; 2) uniaxial strain and biaxial strain have different effects on the band gap of C568; and 3) uniaxial can induce large optical anisotropy, while biaxial strain cannot. In the following sections, these findings are discussed in detail.
The uniaxial strain effect is firstly studied. The electronic band structures with different strain are shown in Fig. 3. When tensile strain is applied, the band structures show indirect character. The locations of the CBM and VBM remain unchanged. Moreover, it can be clearly seen that with increasing tensile strain, the band gap decreases. Under a 10% strain, the band gap is 0.630 eV smaller than that of the equilibrium state. The situation is different when compressive strain is applied. Under a small strain (–2%), the band gap is still indirect. However, when the strain is increased to –6%, the band gap become direct. While the VBM position remains at the M point, the CBM changes from the Γ to the M point. When compressive strain is further enhanced, the direct band structures are still kept. For the direct band gap, the optical transition between VBM and CBM becomes efficient. Electrons can be excited from VBM to CBM without the assistance of lattice phonons, which is very beneficial to the optoelectronic applications. To clearly show the change of band gap, in Fig. 4, we present the evolution of band gap with external uniaxial strain. It can be seen that the band gap of C568 decreases with the increasing tensile strain, and the indirect character is preserved. When compressive strain increase from 0 to –4%, the band gap increases, however, it begins to decrease when the compressive strain exceeds –4%, and becomes direct as discussed above.
Figure 3.(Color online) Band structures of C568 with different uniaxial strains, with the Fermi level set at zero and marked with the black dashed line. Positive and negative values of strains indicate tensile and compressive cases, respectively.
Figure 4.(Color online) (a) Band gap, (b) band edge position, (c) optical absorption, and (d) effective mass of C568 with different uniaxial strains. CBM_Γ and CBM_M are the conduction band minimum at Γ and M points. The vacuum level is taken as the zero energy reference in (b).
The transition from the indirect to direct band gap can be understood from the change of band edge positions at different k points with the applications of strain, as shown in Fig. 4(b). The absolute band edge position is calculated through setting the vacuum level (Evac) as the zero energy reference of calculation[
The difference responses lead to a crossover for the CBM_M and CBM_Γ when the compressive strain increases to –4%. Thus, the indirect band structures transform to direct. Moreover, under compressive strain, due to the energy decrease of the CBM_M state, the band gap value decreases when compressive strain is at the range of –4% to –10%.
The optical anisotropy is also a critical factor in design of polarized optoelectronic devices[
In the following part, the effect of biaxial strain is investigated. We have firstly calculated the band structures of C568 with different biaxial strain, which are shown in Fig. 5. For the case of tensile strain, the evaluation of band structures is similar to the case of uniaxial strain. The indirect gap character is intact, and the gap value is smaller under larger stain.
Figure 5.(Color online) Band structures of C568 with different biaxial strains, with the Fermi level set at zero and marked with blacked dashed line. Positive and negative values of strains indicate tensile and compressive cases, respectively.
For compressive biaxial strain, the band structures show some difference with the case of uniaxial strain. When compressive biaxial strain increases to –6%, the direct band structures can be obtained, with the both CBM and VBM at the M point. When the compressive strain increases to –10%, the direct band structures become indirect again. Another point can be found that the biaxial does not lift the double degeneracy of CBM at the M point. However, in the case of uniaxial strain, the degeneracy is lifted due to the breaking of symmetry along the two lattice vectors. Thus, the optical transition may be different in the case of uniaxial and biaxial strain.
In Fig. 6, we have also calculated the band gap with different biaxial strain. It can be seen that the change of band gap is different from the case of uniaxial strain. When tensile strain is exerted to C568, the band gap keeps dropping. As the tensile strain is enlarged from 0 to 10%, the band gap decreases from 1.062 to 0.380 eV. The trend is the same in the case of uniaxial strain.
Figure 6.(Color online) (a) Band gap, (b) band alignment, (c) optical absorption, and (d) effective mass of C568 with different biaxial strains. CBM_Γ and CBM_M are the conduction band minimum at Γ and M points. The vacuum level is taken as the zero-energy reference in (b).
However, applying compressive strain from 0 to –10% can make the band gap increase from 1.062 to 1.612 eV, which is different from the case of uniaxial strain. However, applying compressive strain from 0 to –10% can make the band gap increase from 1.062 to 1.612 eV, which is different from the case of uniaxial strain. The tunable range of band gap is 0.381–1.612 eV, which can cover the near-infrared region and visible light region. It indicates that, in the experiment, applying biaxial strain can be effective to control the band gap of C568. It is suitable for the design of optoelectronic devices which can work at near-infrared light or visible light. Further, the band gap keeps direct in the strain range of –4% to –9%, which is desired for the optical performance. To understand the transition from indirect to direct band gap, we also calculate the band edge positions, considering the CBM at the M and the Γ points, as well as the VBM at the M point. We can see from Fig. 6(b) that the CBM_M shifts downwards with increasing compressive strain, and becomes lower than CBM_Γ in the range from –4% to –9%, indicating the conduction band minimum changes from the M to the Γ points. The downward shift is also consistent with the bonding character of CBM_M. Thus, the direct band gap is formed. In addition, in the direct gap region the VBM also shift down with a slightly larger rate compared with the CBM_M, resulting in the tiny increase of the band gap value with enhanced strain. The optical absorption for C568 with the strain of –6% is also calculated in Fig. 6(c). The results show that the absorption is isotropic along the x and y direction, which is different from the optical anisotropy in case of uniaxial strain. It is because uniaxial strain makes the lattice become rectangle. Compared to the square lattice caused by biaxial strain, the lattice symmetry decreases in the rectangle lattice. The decrease of lattice symmetry along x and y direction renders the optical anisotropy along x and y direction. The effective mass of hole and electron with the external biaxial strain is also calculated (see Fig. 6(d)). Compared to the effective mass with uniaxial strain, the effective mass for electron and hole is relatively larger.
In experiments, some flexible materials can be the ideal substrate for the growth of 2D C568, such as graphene and plastic substrates. Compressive and tensile strains can be applied on C568 by twisting and stretching the substrate. Previous works show that using flexible PVA substrate to encapsulate monolayer 2D material, and twisting soft polymeric substrates with materials are practical ways to modulate the electronic properties of materials and design flexible devices[
4. Conclusion
Through first-principle calculations, the effect of the external strain on the electronic properties of C568 have been investigated theoretically. The calculations show that while in-plane uniaxial and biaxial strains both reduces the band gap of C568 in case of tensile strain, their effects are quite different in case of compressive strain. With increasing compressive uniaxial strain, the band gap of C568 first increases, and then dramatically decreases. In contrast, the application of compressive biaxial strain up to –10% only leads to a slight increase of band gap. Moreover, an indirect-to-direct gap transition can be realized under both types of compressive strain. The results also show that the optical anisotropy of C 568 can be induced under uniaxial strain, while biaxial strain does not cause such an effect. These results could be helpful to experimentally modulate the electronic properties of C568-based nanodevices.
Acknowledgements
This work was supported by NSAF (Grant No. U1930402). Computational resources were provided by Tianhe2-JK at CSRC.
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