2D Octagon-Structure Carbon and Its Polarization Resolved Raman Spectra

Abstract

We predict a new phase of two-dimensional carbon with density functional theory (DFT). It was found to be semimetal with two Dirac points. The vibrational properties and the polarization resolved Raman spectra of the carbon monolayer are predicted. There are five Raman active modes: 574 cm⁻¹ (E_g), 1112 cm⁻¹ (B_1g), 1186 cm⁻¹ (B_2g), 1605 cm⁻¹ (B_2g) and 1734 cm⁻¹ (A_1g). We consider the incident light wave vector to be perpendicular and parallel to the plane of the carbon monolayer. By calculating Raman tensor of each Raman active mode, we obtained polarization angle dependent Raman intensities. Our results will help materials scientists to identify the existence and orientation of octagon-structure carbon monolayer when they are growing it.

1. Introduction

Graphene has become a hot scientific topic since 2004 [1]. A lot of theoretical and experimental results show that graphene has remarkable properties in mechanics, thermology, electronics and optics [2,3]. In recent years, people found many 2D materials from allotropes of carbon and from group V elements. Some of them have potential application in optoelectronic, nano electronic devices and hydrogen storage, because of their unique electronic and mechanical properties [4,5,6,7,8,9,10]. In 2018, Cao et al. found the twisted bilayer graphene is able to be an insulator or superconductor [11,12]. In the last few decades, scientists have sought and researched some other 2D materials for nanoelectronic, optoelectronic and thermoelectronic devices [13], such as hexagonal boron nitride (h-BN) [14,15,16] and transition metal dichalcogenides (TMDCs) [17,18,19,20,21,22]. Zhang et al. predicted a novel 2D octagon-structure monolayer of nitrogen [23]. Its structure is composed of squares and octagons. There are many good electronic properties of this 2D material. It could be a potential candidate for use in the semiconductor devices, spintronics and quantum computation. Octagon-structure materials can be expected to be a good candidate for hydrogen storage. It is a possible high-temperature superconductor [24]. In 2015, Lu et al. fabricated free-standing octagon-shaped carbon nanofiber [25]. In 2017, Zhong’s research group fabricated graphene-like nanoribbons which are periodically embedded with four- and eight-membered rings [26].

In this work, we theoretically predict a new phase of 2D carbon which consists of octagon carbon rings. Raman spectroscopy is a widely used experimental tool to characterize materials without destroying the samples [27,28]. People observed polarization dependent Raman intensity in many 2D materials, such as WS₂ and MoS₂ [29,30,31,32]_. Therefore, we investigated the vibrational properties and predict the Raman spectra of this 2D octagon-structure carbon by density functional theory (DFT) to help materials scientists to identify the existence of this new phase of 2D carbon.

In 2011, a carbon allotrope named as T-carbon was predicted by Su and his co-workers in theory [33]. Six year later, Zhang et al. synthesized T-carbon from pseudo-topotactic conversion of a multi-walled carbon nanotube suspension in methanol by picosecond pulsed-laser irradiation [34]. Therefore, the 2D Octagon-structure Carbon is holds much hope for synthesis in the future. This work sets a goal for materials scientists, providing a way for them to verify their experimental results, and providing a vast playground for theorists.

2. Materials and Methods

Geometry optimization, force constant calculation and dielectric tensor calculation were performed with the Vienna Ab initio simulation package (VASP, version 5.4.4., Wien, Austria) [35]. Phonon wavenumber and phonon modes were obtained with the Phonopy code [36]. The electron-core interactions were treated in the projector augmented wave (PAW) approximation [35,37,38,39]. The density functional was treated by generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) exchange correlation potential [40]. Kinetic energy cutoffs of 520 eV and 400 eV were used in geometry optimization and other calculations, respectively. The k point mesh was 15 × 15 × 1, 20 × 20 × 1 and 51 × 51 × 1 in geometry optimization, force constant calculation and dielectric function calculation, respectively. Vacuum slabs of 5 nm thick were inserted between neighboring 2D atom sheets. The convergence tolerance for the total energy and force calculations were set to 10⁻⁸ eV and 10⁻⁵ eV/ Å, respectively.

The Raman tensor was obtained with the finite displacement method [31,32,41,42,43,44,45,46,47,48,49,50,51,52]:

$$R_{\alpha \beta }\left(j\right)=V{\displaystyle \sum }_{\mu =1}^N{\displaystyle \sum }_{l=1}^3\frac{\partial {\chi }_{\alpha \beta }}{\partial r_l\left(\mu \right)}\frac{e_l^j\left(\mu \right)}{\sqrt{M_{\mu }}}$$

(1)

where V is the volume of unit cell, M_μ is the atomic mass of atom μ, $\partial$χ_αβ/$\partial$r_l(μ) is the first derivative of the electric polarizability tensor with respect to the atomic displacement. It equals the derivative of dielectric tensor ε_αβ divided by 4π because χ_αβ(ω) = (ε_αβ(ω) − δ_αβ)/4π. ω is the frequency of the laser (electric field). e^j_l(μ) is the eigenvector of the j-th phonon mode at Г point. The dielectric tensor of the structure with finite displacement can be obtained with DFT.

Then Raman intensity can be obtained with [31,32]:

$$\mathrm{I}\propto {\omega }_s^4{|{\stackrel{\rightharpoonup }{e}}_s·\overleftrightarrow{R}·{\stackrel{\rightharpoonup }{e}}_i|}^2$$

(2)

where ${\omega }_s$ is the frequency of scattered light; ${\stackrel{\rightharpoonup }{e}}_i$ and ${\stackrel{\rightharpoonup }{e}}_s$ are electric polarization vectors of incident and scattered light, respectively.

3. Results

The unit cell of monolayer octagon-structure carbon consists of eight atoms (Figure 1a). The lattice constants: a = b = 4.87 Å; the bond length l₁ = 1.373 Å, l₂ = 1.466 Å. The buckled displacement $\Delta z$ is zero (Figure 1b). It belongs to space group P4/mmm (123). The stable lattice structures are shown in Figure 1. Figure 2 shows its band energy structures. It is a semimetal with two Dirac points.

Figure 3a is the phonon dispersion of the carbon monolayer. It does not have imaginary vibrating modes. The octagon-structure carbon monolayer belongs to the D_4h point group, whose Raman active modes are A_1g (in-plane vibration), B_1g (in-plane vibration), B_2g (in-plane vibration) and E_g (out-of-plane vibration). The atomic vibration displacements are shown in Figure 3b. The Γ point wavenumbers of the A_1g mode and B_1g mode are 1734 cm⁻¹ and 1112 cm⁻¹, respectively. The Γ point wavenumbers of the two B_2g modes are 1186 and 1605 cm⁻¹. The Γ point wavenumber of the E_g mode is 574 cm⁻¹.

In order to calculate the Raman intensity, we chose three commonly used laser lines whose wavelengths are 488 nm (2.54 eV), 532 nm (2.33 eV) and 633 nm (1.96 eV) [27,30]. The coordinate system is chosen so that the monolayer plane is in the x-y plane. The calculated results show that the Raman tensors agree with the point group D_4h and their irreducible representations. They are the following matrix forms:

$$\begin{gathered} \tilde{R}(A_{1g})=\left(\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& b\end{array}\right) \\ \tilde{R}(B_{1g})=\left(\begin{array}{ccc}c& 0& 0\\ 0& -c& 0\\ 0& 0& 0\end{array}\right) \\ \tilde{R}(B_{2g})=\left(\begin{array}{ccc}0& d& 0\\ d& 0& 0\\ 0& 0& 0\end{array}\right) \\ \tilde{R}(E_g)=\left(\begin{array}{ccc}0& 0& e\\ 0& 0& f\\ e& f& 0\end{array}\right) \end{gathered}$$

(3)

where a, b, c, d, e, f are complex numbers.

When the incident light wave vector is perpendicular to the monolayer plane (x-y plane), the polarization direction of the incident light is in the x-y plane and can be written as $e_i=(\cos \theta ,\sin \theta ,0)$. We considered two polarization directions of scattered light which are parallel and perpendicular to the polarization direction of the incident light, called parallel polarization configuration and perpendicular polarization configuration, respectively. Then the polarization directions of the scattered light can be written as $e_s^{\parallel }=(\cos \theta ,\sin \theta ,0)$ and $e_s^{\perp }=(-\sin \theta ,\cos \theta ,0)$, respectively. Based on the Raman tensors matrix and polarization directions of the incident and scattered light, we can obtain the polarization angle dependent Raman intensity for each mode. The Raman intensities for these polarization configurations are:

$$I^{\parallel }\left(A_{1g}\right)\propto {\left|a\right|}^2$$

(4)

$$I^{\parallel }\left(B_{1g}\right)\propto {\left|c\right|}^2{\cos }^2(2\theta )$$

(5)

$$I^{\parallel }\left(B_{2g}\right)\propto {\left|d\right|}^2{\sin }^2\left(2\theta \right)$$

(6)

$$I^{\parallel }\left(E_g\right)\propto 0$$

(7)

$$I^{\perp }\left(A_{1g}\right)\propto 0$$

(8)

$$I^{\perp }\left(B_{1g}\right)\propto {\left|c\right|}^2{\sin }^2\left(2\theta \right)$$

(9)

$$I^{\perp }\left(B_{2g}\right)\propto {\left|d\right|}^2{\cos }^2\left(2\theta \right)$$

(10)

$$I^{\perp }\left(E_g\right)\propto 0$$

(11)

For the A_1g mode, Equations (4) and (8) mean that the scattered lights intensity is independent of the polarization angle $\theta$. The intensity is zero for the E_g mode. Only the B_1g modes and B_2g modes show the polarization angle dependent intensity. The polarization dependent Raman intensities of the B_1g modes and B_2g modes of the carbon monolayer for three laser lines 488 nm, 532 nm and 633 nm, when the wave vector of the laser is perpendicular to the 2D plane, are shown in Figure 4 (parallel polarization configuration) and Figure 5 (perpendicular polarization configuration). The order of magnitude of Raman intensities excited by different laser lines is different because the Raman intensity is proportional to ${\omega }_s^4$ (Equation (2)). Therefore, we plot ${|{\stackrel{\rightharpoonup }{e}}_s·\overleftrightarrow{R}·{\stackrel{\rightharpoonup }{e}}_i|}^2$ for clarity.

From Figure 4, Raman intensities of the B_1g and B_2g modes in parallel polarization configurations have four maxima and four minima for all the laser lines. Due to the factor $co{s}^2\left(2\theta \right)$ in Equation (5), the maxima of Raman intensities of B_1g modes (Figure 4a) locate at θ = 0° + n × 90° (n = 0, 1, 2, 3); the four minima locate at θ = 45° + n × 90° (n = 0, 1, 2, 3). Due to the factor $si{n}^2\left(2\theta \right)$ in Equation (6), the maxima of Raman intensities of B_2g modes (Figure 4b,c) locate at θ = 45° + n × 90° (n = 0, 1, 2, 3); the four minima locate at θ = 0° + n × 90° (n = 0, 1, 2, 3). The location of the maxima and minima in Figure 5 are result from a similar reason to that in Figure 4. The polarization angle dependent Raman intensity with the incident light wave vector in the z direction is different from some other isotropic 2D materials (e.g., β-Arsenic monolayer [47] and MXenes Zr₂X(X = C and N) [49]). The anisotropic vibrations are responsible for the polarization dependent Raman intensity.

If the incident light wave vector is along the x-axis, the polarization direction of the incident light is in the y-z plane and can be written as $e_i=(0,\cos \theta ,\sin \theta )$. We also considered parallel polarization configuration and perpendicular polarization configuration, in which the polarization directions of the scattered light can be written as $e_s^{\left|\right|}=(0,\cos \theta ,\sin \theta )$ and $e_s^{\perp }=(0,-\sin \theta ,\cos \theta )$, respectively. Based on the Raman tensors matrix and polarization directions of the incident light and scattered light, we can obtain the polarization angle dependent Raman intensity for each mode. The Raman intensities are:

$$I^{\parallel }\left(A_{1g}\right)\propto {\left|a\right|}^2{\cos }^4\theta +{\left|b\right|}^2{\sin }^4\theta +\mathrm{Re}\left(a·b^{*}\right){\sin }^2\left(2\theta \right)/2$$

(12)

$$I^{\parallel }\left(B_{1g}\right)\propto {\left|c\right|}^2{\cos }^4\theta$$

(13)

$$I^{\parallel }\left(B_{2g}\right)\propto 0$$

(14)

$$I^{\parallel }\left(E_g\right)\propto {\left|f\right|}^2{\sin }^2\left(2\theta \right)$$

(15)

$$I^{\perp }\left(A_{1g}\right)\propto \left[{\left|a\right|}^2+{\left|b\right|}^2-2\mathrm{Re}(a·b^{*})\right]{\sin }^2\left(2\theta \right)/4$$

(16)

$$I^{\perp }\left(B_{1g}\right)\propto {\left|c\right|}^2{\sin }^2\left(2\theta \right)/4$$

(17)

$$I^{\perp }\left(B_{2g}\right)\propto 0$$

(18)

$$I^{\perp }\left(E_g\right)\propto {\left|f\right|}^2{\cos }^2\left(2\theta \right)$$

(19)

The angular dependent Raman intensities of different vibrating modes of the carbon monolayer for three laser lines 488 nm, 532 nm and 633 nm, when the wave vector is parallel to the 2D plane, are shown in Figure 6 (parallel polarization configuration) and Figure 7 (perpendicular polarization configuration).

From Figure 6, the maxima locate at 0° and 180° for the A_1g modes (Figure 6a), because |a| is always much larger than |b|, which may change with laser photon energy; the maxima locate at 0° and 180° for B_1g modes (Figure 6b) due to the factor $co{s}^4\theta$; the Raman intensities have four maxima which locate at θ = 45° + n × 90° (n = 0, 1, 2, 3) for the E_g mode (Figure 6c) due to the factor $si{n}^2\left(2\theta \right)$.

From Figure 7a,b, Raman intensities of the A_1g and B_1g modes in perpendicular polarization configurations have four maxima for all the laser lines; the maxima intensities locate at θ = 45° + n × 90° (n = 0, 1, 2, 3) and the minima ones are zero at θ = 0° + n × 90° (n = 0, 1, 2, 3), due to the factor $si{n}^2\left(2\theta \right)$. The Raman intensities have four maxima which locate at θ = 0° + n × 90° (n = 0, 1, 2, 3) for the E_g mode (Figure 7c) due to the factor $co{s}^2\left(2\theta \right)$.

By integrating the Raman intensity of all polarization directions, we obtained the monolayer’s nonpolarized Raman intensity of the monolayer when the incident light wave vector is the along z and x direction (Figure 8). The factor ${\omega }_s^4$ in Equation (2) is omitted in Figure 8 for clarity.

When the incident light is along the z-axis, the monolayer has four distinct peaks (Figure 8a) at 1112 cm⁻¹ (B_1g), 1186 cm⁻¹ (B_2g), 1605 cm⁻¹ (B_2g) and 1734 cm⁻¹ (A_1g). When the incident light is along the x-axis, the monolayer has three peaks (Figure 8b) at 574 cm⁻¹ (E_g), 1112 cm⁻¹ (B_1g) and 1734 cm⁻¹ (A_1g). Whatever the direction of the incident light, the Raman intensity of 1112 cm⁻¹ (B_1g) and 1734 cm⁻¹ (A_1g) are always the strongest and the second strongest.

We calculated the energy of the octagon 2D carbon and compare it with the other allotrope: graphene, graphite and diamond (

Table 1. Free energy of graphene, graphite, diamond and octagon carbon.

Material	Average Energy/Atom
Graphene	−9.224 eV
Octagon Carbon monolayer	−8.711 eV
Graphite	−9.306 eV
Diamond	−9.099 eV
Octagon Carbon Bulk	−8.782 eV

). It is found that the octagon carbon monolayer’s energy is greater than graphene and diamond. It means graphene, graphite and diamond are more stable than the octagon 2D carbon.

4. Discussion and Conclusions

The Raman shifts for graphene and graphite are 1350, 1600 and 2700 cm⁻¹ [53,54]; the Raman shifts for diamond are 1332, 2015 and 2176 cm⁻¹ [55]. The strong and distinct 1112 cm⁻¹ (B_1g) and 1734 cm⁻¹ (A_1g) Raman shifts of octogen 2D carbon are unique. They can help materials scientists to distinguish it from graphene, graphite and diamond.

There are five Raman active modes: 574 cm⁻¹ (E_g), 1112 cm⁻¹ (B_1g), 1186 cm⁻¹ (B_2g), 1605 cm⁻¹ (B_2g) and 1734 cm⁻¹ (A_1g). When the incident light wave vector is perpendicular to the plane of the carbon monolayer, B_1g and B_2g modes can show polarization-angle-dependent Raman intensity. This is different from some other isotropic 2D materials (e.g., β-Arsenic monolayer [47], Zr₂C monolayer and Zr₂N monolayer [49]). There all are four maxima and minima for all the laser lines when the polarization direction of the scattered light is parallel or perpendicular to that of the incident light. The incident light wave vector is parallel to the plane of the carbon monolayer. For the A_1g and B_1g modes, Raman intensities have two (four) minima and two (four) maxima when the polarization direction of scattered light is parallel (perpendicular) to that of the incident light. The intensity is zero for the B_2g mode. For the E_g mode, there always are four minima and four maxima when the polarization direction of the scattered light is parallel or perpendicular to that of the incident light.

The Dirac cone may result in many excellent physical properties, such as very high carrier mobility and an anomalous quantum Hall effect. Due to the two-phonon double resonance Raman process between Dirac points, there will be some double resonant bands whose Raman shifts are approximately twice the Γ point phonon frequencies [54,56].

In conclusion, we predicted a new phase of 2D carbon: 2D octagon-structure carbon. It is found to be semimetal with two Dirac points. We investigated its vibrational properties and polarization resolved Raman spectra excited by three commonly used laser lines 488nm, 532nm and 633nm. The Raman spectra has four and three distinct peaks when the incident light wave vector is perpendicular and parallel to the carbon monolayer, respectively. We expect this octagon-structure carbon can be grown in experiments and would have wide applications.