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Article

Study of the Electronic Band Structure and Structural Stability of Al(CN)2 and Si(CN)2 by Density Functional Theory

by
Sok-I Tam
1,
Pak-Kin Leong
2,
Chi-Pui Tang
1,2,*,
Weng-Hang Leong
1,
Toshimori Sekine
3,
Chi-Long Tang
2,
Kuan-Vai Tam
4 and
Kin-Tak U
1,*
1
Faculty of Innovation Engineering, Macau University of Science and Technology, Taipa, Macao 999078, China
2
State Key Laboratory of Lunar and Planetary Sciences, Macau University of Science and Technology, Taipa, Macao 999078, China
3
Center for High Pressure Science and Technology Advanced Research, Shanghai 201203, China
4
School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China
*
Authors to whom correspondence should be addressed.
Crystals 2023, 13(5), 824; https://doi.org/10.3390/cryst13050824
Submission received: 31 March 2023 / Revised: 6 May 2023 / Accepted: 10 May 2023 / Published: 16 May 2023
(This article belongs to the Special Issue Density Functional Theory (DFT) and Beyond for Crystalline Materials)
Browse Figures
Versions Notes

Abstract

:
By substituting the A site in P 2 1 / c - A ( C N ) 2 and varying the lattice parameters a, b, c, and the unit-cell angles, along with using crystal graph convolutional neural networks to calculate their cohesive energy, the candidate compounds, A l ( C N ) 2 and S i ( C N ) 2 , were selected from the structure with the lowest cohesive energy. The two candidate structures were then optimized using first-principles calculations, and their phonon, electronic, and elastic properties were computed. As a result, two dynamically stable structures were found: A l ( C N ) 2 with a space group of C m c m and S i ( C N ) 2 with a space group of R 3 ¯ m . Their phonon spectra exhibited no imaginary frequencies; thus, their elastic constants satisfied the mechanical stability criteria. Structurally, S i ( C N ) 2 is similar to 6H- S i C and 15R- SiC . Its elastic constants indicated that it is harder than those SiC materials. A l ( C N ) 2 exhibits metallic properties and the indirect wide-bandgap of S i ( C N ) 2 was calculated by the generalized gradient approximation, the local density approximation, and the screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06) is found to be 3.093, 3.048, and 4.589 eV, respectively. According to this wide bandgap, we can conclude that S i ( C N ) 2 has the potential to be used in high-temperature and high-power environments, making it usable in a broad range of applications.

1. Introduction

Carbon and nitrogen are the primary elements of planetary atmospheres, and anion cyanide or isocyanide has numerous potential applications in industrial materials and catalysts [1,2,3]. Previous work has reported the stability and electronic properties of six structures ( P 4 ¯ 3 m , P 4 2 n m , R 3 m , P 2 1 / c , R 3 ¯ m , and C 2 / m ) of alkaline-earth metal cyanides A ( C N ) 2 (A = Be , Mg , Ca , Sr , and Ba ) using modified crystals of known structures and first-principles calculations [4]. The calculations demonstrate that some structures are potentially stable, including the C 2 / m structure, which is metallic, and other structures with bandgaps ranging from 2.83 to 6.33 eV.
These A ( C N ) 2 compound variants can be applied to systems other than alkaline-earth elements. The bonding nature between C and N in such systems may alter the structure, properties, and stability even when the formula is the same. The technique of structural simulation is the most popular and useful way to investigate valuable components made of well-known elements. Some reports of amorphous materials with various C/N ratios have been developed (e.g., Matsunaga et al. [5]). However, there have been relatively few studies on compounds in regard to stoichiometry, such as A ( C N ) 2 , where A is not a common + 2 valent element (e.g., alkaline-earth metal and zinc). Therefore, it is worthwhile to investigate the potential structures of these compounds using computational simulations.
Recently, crystal graph convolutional neural networks (CGCNN) have been used to predict material properties, receiving significant attention in materials science [6,7,8,9]. CGCNN is exceptional for predicting properties such as formation energy, bandgap, Fermi energy, and elastic properties [6]. Crystal frameworks are represented by graphs while graph neural networks are used in forecast material performance.
In this study, the chosen structure is based on the earlier report of the structure of P 2 1 / c alkaline-earth metal cyanide A ( C N ) 2 [4]. Thousands of unit cells are generated by combining various side lengths and unit-cell angles, and the A atoms in A ( C N ) 2 are replaced with other elements. We then use CGCNN to calculate their cohesive energy to find a structure with the lowest energy. This structure is considered to be the most advantageous configuration. Other than alkaline-earth metals, aluminum and silicon were chosen as A in the A ( C N ) 2 formula because they have the lowest cohesive energy [10]. Additionally, their carbon and nitrogen compounds have numerous essential properties.
In the Al-C-N system, A l ( C N ) 3 has been extensively used as a catalyst (e.g., in the textile industry) and for porous materials [1]. A l ( C N ) x ( x = 1 , 2 , 3 ) is investigated for applications in materials and catalytic reactions [11]. Additionally, molecules such as A l NC , A l C N , and AlC n N ( n = 1 , 2 ) may exist in the carbon star IRC+10216 [12,13]; therefore, their related structures, electronic properties, and optical properties have been studied.
Materials such as S i C N and SiC x N y (where x and y are any nonzero numbers) have been widely studied for their high hardness [14,15], low brittleness, low thermal expansion, high chemical, and thermal stability in the Si-C-N system. As a result, such materials are used as thin film materials, photoelectric materials, electromagnetic-wave absorption materials, etc.

2. Materials and Methods

In this work, we would like to predict the cohesive energy of A ( C N ) 2 . Thus, we used a CGCNN trained by inorganic crystal material data from the Materials Project [16]. The chosen structures are based on an earlier report of the structure of P 2 1 / c alkaline-earth metal cyanide A ( C N ) 2 [4]. We generate 1050 unit cells with varying side lengths and unit-cell angles, then substitute the A atoms in A ( C N ) 2 with other elements. The replaced elements include a total of 35 elements from the second to sixth periods of the first to seventh main groups. We then use CGCNN to determine the cohesive energy of a total of 36,750 unit cells [10]. Finally, aluminum and silicon (i.e., A l ( C N ) 2 and S i ( C N ) 2 ) were chosen for further structural optimization by using density functional theory (DFT) calculations, because they are the elements with the lowest cohesive energy besides alkaline-earth metals. In the following sections, we examine the properties (e.g., phonon, electronic, elastic) of A l ( C N ) 2 and S i ( C N ) 2 using DFT.
First-principles calculations based on DFT [17,18] were implemented in the Cambridge Sequential Total Energy Package (CASTEP) [19,20]. For the exchange-correlation function, a generalized gradient approximation (GGA) in the form of the Perdew–Burke–Ernzerhof (PBE) [21] and the local density approximation (LDA) [22,23] were considered using the on-the-fly-generated ultrasoft pseudopotential method [24]. The plane-wave kinetic-energy cutoff was set to 570 eV and, to achieve structural optimization, the k-points separation was set to 0.07 Å − 1 for all calculations in the Brillouin zone, with the Monkhorst–Pack method applied for the point distribution [25]. Since the GGA and LDA underestimate the energy bandgap [21], we used the screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06) [26] as a reference to calculate A l ( C N ) 2 and S i ( C N ) 2 along with an on-the-fly-generated norm-conserving pseudopotential [27]. The plane-wave kinetic-energy cutoff was set to 990.0 eV for HSE06. The minimization algorithm we used was the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method [28] with convergence tolerances set to 5.0 × 10 − 6 eV/atom for energy, 0.01 eV/Å for maximum force, 0.02 GPa for maximum stress, and 5.0 × 10 − 4 Å for maximum displacement. The phonon spectra were calculated by using the finite-displacement method.
All the results of LDA and HSE06, including lattice parameters and band structure, are shown in the Supplementary Materials. Based on the prior reports [4], the GGA functional is more appropriate for this system. Therefore, the GGA functional is used in the subsequent calculations.
We also preformed a Mulliken population analysis for A l ( C N ) 2 and S i ( C N ) 2 and the elastic properties of A l ( C N ) 2 , S i ( C N ) 2 , 6H- S i C , and 15R- S i C . The graphical displays in this work are generated with BIOVIA Materials Studio.

3. Results and Discussion

3.1. Structures of A l ( C N ) 2 and S i ( C N ) 2

After searching with CGCNN for the structure with the lowest energy from among a large number of structures, we made use of DFT for the full relaxation geometry optimization. The structural symmetries of A l ( C N ) 2 and S i ( C N ) 2 are C m c m and R 3 ¯ m , respectively. The unit-cell structure diagrams are shown in Figure 1 and the lattice parameters are listed in Table 1.
The C m c m structure of A l ( C N ) 2 is shown in Figure 1a; the coordination numbers of aluminum, nitrogen, and carbon are 6, 4, and 3, respectively. The bond angles ∠CCC and ∠CCN are both near to 120° (Table 1), indicating that the carbon in A l ( C N ) 2 is s p 2 hybridized and conjugated. This carbon chain structure is similar to the previously C 2 / m structure of B e ( C N ) 2 and M g ( C N ) 2 in (Figure 1c,d) [4].
In addition to the similar bond angles ∠CCC and ∠CCN in Table 1, the C − C bond length of A l ( C N ) 2 is comparable to that of B e ( C N ) 2 and M g ( C N ) (1.520 Å vs. 1.501 Å and 1.598 Å, respectively). It should be noted that these three compounds are conductive and dynamically stable. The difference between them is that the C − N bond length in A l ( C N ) 2 is longer than that of B e ( C N ) 2 and M g ( C N ) 2 (1.343 Å vs. 1.279 Å and 1.270 Å, respectively). It is possible that aluminum has one extra valence electron than beryllium and magnesium, and the nitrogen connected to the metal transforms from C ( s p 2 ) =N ( s p 2 ) bond (e.g., vinylidendiimine H N = C = C = N H , d C − N = 1.27 Å [29]) to C ( s p 2 ) − N ( s p 3 ) bond.
In terms of symmetry and the unit-cell diagram, the R 3 ¯ m structure of S i ( C N ) 2 is very close to the reported R 3 ¯ m structure of S r ( C N ) 2 and B a ( C N ) 2 , as shown in Figure 1b,e. The coordination numbers of silicon, nitrogen, and carbon are 6, 4, and 4, respectively. In contrast, the coordination numbers of strontium/barium, nitrogen, and carbon are 8, 4, and 2, respectively. Figure S1 in the Supplementary Materials displays the rhombohedral representation of these structures. In S i ( C N ) 2 , the C N group is surrounded by four silicon atoms, and each C N group is close to forming a C − C single bond (1.645 Å in Table 1). Meanwhile, it is considered that there is no chemical bond between the carbon atoms in S r ( C N ) 2 and B a ( C N ) 2 because the shortest distance between two carbons atoms is 2.903 and 3.069 Å, respectively.
The length of the unit-cell edge of S i ( C N ) 2 is a = 5.672 Å, which is close to that of S r ( C N ) 2 and B a ( C N ) 2 (5.438 and 5.749 Å, respectively). However, the unit-cell angle of S i ( C N ) 2 is α = 27 . 51 ∘ and the cell volume is V = 34.35 Å 3 , which is significantly smaller than those of S r ( C N ) 2 ( α = 52 . 06 ∘ , V = 95.05 Å 3 ) and B a ( C N ) 2 ( α = 51 . 69 ∘ , V = 108.1 Å 3 ). These differences might be attributed to the fact that silicon has a much smaller atomic radius (1.11 Å) than strontium (2.19 Å) and barium (2.53 Å) [30], resulting in not enough space between silicon atoms to separate the C N groups. In addition, the shortest distance between carbon and silicon in S i ( C N ) 2 is 2.912 Å (Table 1), which is much greater than the sum of the atomic radii of these two elements ( 0.67 + 1.11 = 1.78 Å ), indicating that they do not form bonds.
The conventional cell of A l ( C N ) 2 and the hexagonal representation of S i ( C N ) 2 are found to be similar and are arranged layer by layer in the sequence of ⋯ A − N − C − N − A ⋯ (Figure 1a,b). The difference is the type of hybridization of carbon atoms. As observed in the lattice parameters of S i ( C N ) 2 in Table 1, although the C − C bond (1.645 Å) has a slightly longer bond length than the normal C − C single bond (1.52–1.58 Å) [31], and the C − N bond length (1.468 Å) is much longer than the general cyano carbon-nitrogen triple bond C ≡ N ( 1.17 Å ) , the bond angles ∠CCC, ∠CCN, ∠CNA, and ∠ANA are both close to the sp3 hybrid bond angle of 109°. It is determined that all of the carbon and nitrogen atoms in Si(CN)2 are sp3 hybridized. When combined with the previously described C(sp2)−N(sp3) bond in A l ( C N ) 2 , there are four types of C − N bonds, and the order of bond length from long to short is related to the degree of hybridization: (1) Both carbon and nitrogen are s p 3 hybridized C( s p 3 ) − N ( s p 3 ) such as R 3 ¯ m - S i ( C N ) 2 (1.468 Å); (2) carbon is s p 2 hybridized and nitrogen is s p 3 C( s p 2 ) − N ( s p 3 ) such as C m c m - A l ( C N ) 2 (1.343 Å); (3) both atoms are s p 2 hybridized C( s p 2 ) − N ( s p 2 ) such as C 2 / m - B e ( C N ) 2 and M g ( C N ) 2 (1.279 and 1.270 Å, respectively) [4]; (4) both atoms are s p hybridized C( s p ) − N ( s p ) such as R 3 ¯ m - S r ( C N ) 2 and B r ( C N ) 2 (1.182 and 1.183 Å, respectively) [4].
In addition, the R 3 ¯ m structure of S i ( C N ) 2 is quite similar to that of 6H- and 15R- S i C . S i ( C N ) 2 is comparable to 6H- S i C when four hexagons are used as reference points (see Figure 2), while the c / a ratio of 15R- S i C is roughly twice that of S i ( C N ) 2 . For both 6H- and 15R- S i C , as well as S i ( C N ) 2 , all atoms in the structure are hybridized in s p 3 orbitals. For similar structures, S i ( C N ) 2 is believed to exhibit some mechanical properties similar to S i C . Aside from shortening the A side of the lattice, the addition of nitrogen atoms may change some properties. For example, S i C N films have higher scratch-and-wear resistance than S i C [32], and other silicon carbonitrides have properties such as high piezoresistivity [33], different hardness, and an elastic modulus [34].

3.2. Phonon and Electrical Properties

The phonon spectra of A l ( C N ) 2 and S i ( C N ) 2 are shown in Figure 3. There are no imaginary frequencies, indicating that both structures are dynamically stable. As a reference comparison, B e ( C N ) 2 and M g ( C N ) 2 of the C 2 / m structure and S r ( C N ) 2 and B a ( C N ) 2 of the R 3 ¯ m structure also have no imaginary frequencies at zero pressure. For S i ( C N ) 2 , which is also an R 3 ¯ m structure, the lowest frequency of point B in the Brillouin zone of its phonon spectrum is 490 cm − 1 , which is much greater than that of S r ( C N ) 2 at 70 cm − 1 and B a ( C N ) 2 at 65 cm − 1 [4]. Moreover, R 3 ¯ m - M g ( C N ) 2 of the same period has imaginary frequencies. This implies that S i ( C N ) 2 has a higher binding energy and a more stable structure.
According to the band structure shown in Figure 4, A l ( C N ) 2 has metallic properties similar to the C 2 / m structure of B e ( C N ) 2 and M g ( C N ) 2 , which is thought to be caused by the conjugated carbon chain structure. On the other hand, S i ( C N ) 2 has an indirect bandgap of 3.093 eV with the GGA functional (see Table S1 and Figure S1: 3.048 and 4.551 eV for the LDA and HSE06 functionals, respectively). Compared with 4.143 eV for S r ( C N ) 2 and 4.122 eV for B a ( C N ) 2 , the bandgap of S i ( C N ) 2 is smaller. However, it is larger than the 2.01 eV of 6H- S i C [35]. Since the experimental bandgap of 6H- S i C is 3.02 eV [36] and the calculated bandgap using GGA functional is an underestimate, we assume that S i ( C N ) 2 is a wide-bandgap semiconductor.
In the partial density of states (PDOS) of S i ( C N ) 2 and A l ( C N ) 2 , shown in Figure 5, the p electrons of carbon and nitrogen play a key role near the Fermi level in both structures. In A l ( C N ) 2 , the p electrons of nitrogen and carbon mainly compose the band which passes through the Fermi level. In S i ( C N ) 2 , the top of the valence band is mainly composed of p electrons of nitrogen and carbon, whereas the bottom of the conduction band is mainly composed of p electrons of carbon.

3.3. Population Analysis

To further investigate the bonding characteristics of the two compounds, we performed Mulliken population analysis [37], and the results are presented in Table 2. The C − C bond population in A l ( C N ) 2 is 2.05, indicating a strong covalent bond, and the C − N bond population is 0.95, which is also indicative of covalent bonding characteristics [38]. The two A l − N bonds have ionic bond characteristics, with the bond at 2.009 Å having stronger ionic bonding properties. In S i ( C N ) 2 , the C − C bond population is even greater, indicating stronger covalent bonding properties, whereas the covalent nature of the C − N bond is weakened and tends towards ionic bonding properties, and the S i − N bond is covalent. This implies that the lone pair of electrons on the nitrogen atom is donating electrons to the carbon atom, i.e., the nitrogen atom exerts an electron-donating effect on the carbon atom. This increases the electron density of the carbon atom and hence lengthens the C − C single bond.

3.4. Elastic Property

To investigate the potential use of S i ( C N ) 2 as a hard material similar to S i C , we calculated its elastic constants, which are presented in Table 3. Elastic constants of A l ( C N ) 2 , 6H-, and 15R- S i C are also included for comparison. All compounds meet the mechanically stable criteria of the hexagonal system [39]:
C 11 − C 12 > 0 , C 44 > 0 , C 11 + C 12 C 33 − 2 C 13 2 > 0
The results show that the moduli of S i ( C N ) 2 , including the bulk modulus, shear modulus, and Young’s modulus, are significantly greater than those of both 6H- and 15R- S i C . Specifically, the bulk modulus of S i ( C N ) 2 (347.7 GPa) is 163.5% and 165.7% that of 6H- S i C (212.7 GPa) and 15R- S i C (209.9 GPa), respectively. The shear modulus of S i ( C N ) 2 (354.0 GPa) is 190.5% and 192.0% that of 6H- S i C (185.8 GPa) and 15R- S i C (184.4 GPa), respectively. The Young’s modulus of S i ( C N ) 2 (792.8 GPa) is 183.7% and 185.3% that of 6H- S i C (431.6 GPa) and 15R- S i C (427.9 GPa), respectively. These results may be due to the introduction of nitrogen, which leads to a harder material [32,42]. Therefore, S i ( C N ) 2 has the potential to be used in SiC x N y systems as a hard material. It may be a potential substitute for SiC in applications involving high-temperature and high-stress conditions. Furthermore, as shown in Table 3, the elastic constants of A l ( C N ) 2 are mostly smaller than those of S i ( C N ) 2 , which is consistent with the phonon spectra results.

4. Conclusions

We have selected two candidate materials through the calculation of cohesive energies of CGCNN. After the structural optimization by DFT, we consider them to be dynamically stable and mechanically stable based on the calculation of phonon and elastic constant, respectively. The structures and physical properties of C m c m - A l ( C N ) 2 and R 3 ¯ m - S i ( C N ) 2 are also studied.
The C m c m structure of A l ( C N ) 2 is similar to the C 2 / m space group of B e ( C N ) 2 and M g ( C N ) 2 . The R 3 ¯ m structure of S i ( C N ) 2 is similar to that of S r ( C N ) 2 and B a ( C N ) 2 . There are four types of C − N bonds and the order of bond length is related to the degree of hybridization.
The band structure and density of state calculations thus suggest that S i ( C N ) 2 is an indirect wide-bandgap semiconductor with a bandgap of 3.093 eV as per GGA, whereas A l ( C N ) 2 has metallic properties. The p electrons of carbon and nitrogen play a key role near the Fermi level in both structures. Wide-bandgap semiconductors can operate at higher voltages and temperatures with greater efficiency and better thermal stability.
In the comparison of elastic constants, S i ( C N ) 2 is harder than both 6H- S i C and 15R- S i C due to the introduction of nitrogen, so S i ( C N ) 2 may be used as a hard material in SiCxNy systems. It may have high thermal conductivity, chemical stability, and durability in hightemperature and high-pressure environments. Due to its wide bandgap and relatively hard properties, Si(CN)2 may have widespread applications in equipment operating under extreme conditions, such as high-temperature and high-pressure sensors. It thus makes a promising candidate for use in next-generation power devices and high-frequency applications. Just like other wide-bandgap materials such as SiC, GaAs, AlGaN, and GaN, Si(CN)2 has the potential to be used in lasers, LED lighting, and RF signal processing due to its similarly wide bandgap, thus offering the potential for use in a broad range of applications.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/cryst13050824/s1, Figure S1: The unit-cell structure diagrams of (a) primitive cell structure of C m c m - A l ( C N ) 2 , (b) rhombohedral representation of R 3 ¯ m - S i ( C N ) 2 , (c) primitive cell structure of C 2 / m - M g ( C N ) 2 , and (d) rhombohedral representation of R 3 ¯ m - S r ( C N ) 2 and B a ( C N ) 2 . Gray: carbon; Blue: nitrogen; Purple: aluminum; Yellow: silicon; Green: alkaline earth metal; Figure S2: Band structures of A l ( C N ) 2 and S i ( C N ) 2 with different functional; Figure S3: Total density of states and partial density of states of A l ( C N ) 2 and S i ( C N ) 2 with LDA functional; Figure S4: Total density of states and partial density of states of A l ( C N ) 2 and S i ( C N ) 2 with HSE06 functional; Table S1: Lattice parameters of C m c m - A l ( C N ) 2 (shown in primitive cell, Z = 2 ) and R 3 ¯ m - S i ( C N ) 2 (rhombohedral representation, Z = 1 ) with different functional; E coh is the cohesive energy; Table S2: Atomic coordinates of A l ( C N ) 2 and S i ( C N ) 2 ; Table S3: Atomic coordinates of 6H- S i C and 15R- S i C ; Table S4: Mulliken atomic population analysis of the A l ( C N ) 2 and S i ( C N ) 2 .

Author Contributions

Conceptualization, S.-I.T., C.-P.T. and T.S.; methodology, S.-I.T., P.-K.L. and C.-P.T.; software, S.-I.T. and P.-K.L.; validation, C.-P.T. and P.-K.L.; formal analysis S.-I.T., P.-K.L., C.-L.T., W.-H.L. and K.-V.T.; investigation, C.-P.T., T.S. and W.-H.L.; data curation, S.-I.T. and P.-K.L.; writing—original draft preparation, S.-I.T., P.-K.L., C.-P.T. and T.S.; writing—review and editing, S.-I.T., P.-K.L., T.S., C.-P.T., K.-V.T. and K.-T.U.; visualization, S.-I.T., P.-K.L. and C.-L.T.; supervision, C.-P.T. and K.-T.U. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Science and Technology Development Fund (FDCT) of Macau [Grants 0111/2020/A, 0048/2020/A1, 0105/2020/A3, 0014/2022/A1, and 0122/2022/A. File No. SKL-LPS(MUST)-2021-2023], and partially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 41974099).

Data Availability Statement

The data that support the findings of this study are available within the article.

Acknowledgments

We would like to thank Bo-Chi Cha for her assistance in editing the article format.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
CGCNNCrystal Graph Convolutional Neural Network
DFTDensity Functional Theory
CASTEPCambridge Sequential Total Energy Package
GGAGeneralized Gradient Approximation
LDALocal Density Approximation
PDOSPartial Density of States

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Crystals 13 00824 g001 550
Figure 1. The unit-cell structure diagrams of (a) conventional cell structure of C m c m - A l ( C N ) 2 , (b) hexagonal representation of R 3 ¯ m - S i ( C N ) 2 , (c,d) conventional cell structure of C 2 / m - B e ( C N ) 2 and M g ( C N ) 2 , respectively, and (e) hexagonal representation of R 3 ¯ m - S r ( C N ) 2 and B a ( C N ) 2 . Gray: carbon; Blue: nitrogen; Purple: aluminum; Yellow: silicon; Green: alkaline-earth metal.
Figure 1. The unit-cell structure diagrams of (a) conventional cell structure of C m c m - A l ( C N ) 2 , (b) hexagonal representation of R 3 ¯ m - S i ( C N ) 2 , (c,d) conventional cell structure of C 2 / m - B e ( C N ) 2 and M g ( C N ) 2 , respectively, and (e) hexagonal representation of R 3 ¯ m - S r ( C N ) 2 and B a ( C N ) 2 . Gray: carbon; Blue: nitrogen; Purple: aluminum; Yellow: silicon; Green: alkaline-earth metal.
Crystals 13 00824 g001
Crystals 13 00824 g002 550
Figure 2. A top view of the a-axis showing (a): S i ( C N ) 2 , (b): 6H- S i C , and (c): 15R- S i C .
Figure 2. A top view of the a-axis showing (a): S i ( C N ) 2 , (b): 6H- S i C , and (c): 15R- S i C .
Crystals 13 00824 g002
Crystals 13 00824 g003 550
Figure 3. Phonon dispersion and phonon density of states spectra of A l ( C N ) 2 and S i ( C N ) 2 .
Figure 3. Phonon dispersion and phonon density of states spectra of A l ( C N ) 2 and S i ( C N ) 2 .
Crystals 13 00824 g003
Crystals 13 00824 g004 550
Figure 4. Band structures of A l ( C N ) 2 and S i ( C N ) 2 .
Figure 4. Band structures of A l ( C N ) 2 and S i ( C N ) 2 .
Crystals 13 00824 g004
Crystals 13 00824 g005 550
Figure 5. Total density of states and partial density of states of A l ( C N ) 2 and S i ( C N ) 2 .
Figure 5. Total density of states and partial density of states of A l ( C N ) 2 and S i ( C N ) 2 .
Crystals 13 00824 g005
Table
Table 1. Lattice parameters of A ( C N ) 2 (A = A l , B e , M g , S i , S r , and B a ). E coh is the cohesive energy.
Table 1. Lattice parameters of A ( C N ) 2 (A = A l , B e , M g , S i , S r , and B a ). E coh is the cohesive energy.
A A l B e 1 M g 1 S i S r 1 B a 1
Space group 2 C m c m C 2 / m C 2 / m R 3 ¯ m R 3 ¯ m R 3 ¯ m
a = b (Å)6.2985.8993.1745.6725.4385.749
c (Å)2.6454.5315.981 = a = a = a
α = β ( ∘ )90.0082.3282.8827.5152.0651.69
γ ( ∘ )50.7724.7550.61 = α = α = α
V (Å 3 )81.2865.3946.1234.3595.05108.1
d A − C (Å)–––2.912 32.7962.995
d A − N (Å)2.009/2.0231.7412.091/2.2991.8482.8462.999
d C − N (Å)1.3431.2791.2701.4681.1821.183
d C − C (Å)1.5201.5011.5981.6452.900 33.069 3
∠CNA ( ∘ )115.9/132.8133.4121.5/128.7122.6104.4105.3
∠ANA ( ∘ )81.6593.1180.9293.76114.0113.3
∠CCC ( ∘ )120.9114.8116.2110.1––
∠CCN ( ∘ )119.5122.6121.9108.8––
E coh (eV) − 7.841 − 7.835 − 7.448 − 8.323 − 7.656 − 7.739
Bandgap (eV)– 4– 4– 43.0934.1434.122
1 Ref. [4]. 2  C m c m and C 2 / m is represented in primitive cell. R 3 ¯ m is represented in rhombohedral cell. 3 The value represents the shortest distance between two atoms. 4 They are conductive.
Table
Table 2. Mulliken bond population analysis of A l ( C N ) 2 and S i ( C N ) 2 .
Table 2. Mulliken bond population analysis of A l ( C N ) 2 and S i ( C N ) 2 .
A A l S i
BondPopulationLength (Å)PopulationLength (Å)
C − N 0.951.3430.571.468
C − C 2.051.5202.351.645
A − N 0.332.0091.361.848
A − N 0.692.023
Table
Table 3. Elastic constants of A l ( C N ) 2 , S i ( C N ) 2 , 6H- S i C , and 15R- S i C . B is the bulk modulus, G is the shear modulus, and E is Young’s modulus. All units are GPa.
Table 3. Elastic constants of A l ( C N ) 2 , S i ( C N ) 2 , 6H- S i C , and 15R- S i C . B is the bulk modulus, G is the shear modulus, and E is Young’s modulus. All units are GPa.
A l ( C N ) 2 S i ( C N ) 2 6H- S i C 15R- S i C
PresentPresentRef. 1Expt. 2PresentRef. 1Present
C 11 328.0780.8498 501 ± 4 484.3498476.6
C 12 93.5108.789 111 ± 5 104.19997.9
C 13 37.428.848 52 ± 9 51.44950.7
C 33 925.01290.0533 553 ± 4 531.7538537.5
C 44 207.4296.0162 163 ± 4 159.2162157.5
B225.7347.7211 212.7214209.9
G114.8354.0193 185.8191184.4
E294.4792.8444 431.6442427.9
1 DFT calculation with GGA functional from Ref. [40]. 2 Experimental data from Ref. [41].
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Tam, S.-I.; Leong, P.-K.; Tang, C.-P.; Leong, W.-H.; Sekine, T.; Tang, C.-L.; Tam, K.-V.; U, K.-T. Study of the Electronic Band Structure and Structural Stability of Al(CN)2 and Si(CN)2 by Density Functional Theory. Crystals 2023, 13, 824. https://doi.org/10.3390/cryst13050824

AMA Style

Tam S-I, Leong P-K, Tang C-P, Leong W-H, Sekine T, Tang C-L, Tam K-V, U K-T. Study of the Electronic Band Structure and Structural Stability of Al(CN)2 and Si(CN)2 by Density Functional Theory. Crystals. 2023; 13(5):824. https://doi.org/10.3390/cryst13050824

Chicago/Turabian Style

Tam, Sok-I, Pak-Kin Leong, Chi-Pui Tang, Weng-Hang Leong, Toshimori Sekine, Chi-Long Tang, Kuan-Vai Tam, and Kin-Tak U. 2023. "Study of the Electronic Band Structure and Structural Stability of Al(CN)2 and Si(CN)2 by Density Functional Theory" Crystals 13, no. 5: 824. https://doi.org/10.3390/cryst13050824

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