Interfacial Stabilities, Electronic Properties and Interfacial Fracture Mechanism of 6H-SiC Reinforced Copper Matrix studied by the First Principles Method

: The interfacial mechanics and electrical properties of the SiC reinforced copper matrix composites were studied via the first principles method. The work of adhesion ( W ad ) and the interfacial energies were calculated to evaluate the stabilities of the SiC/Cu interfacial models. The carbon terminated (CT)-SiC/Cu interfaces were predicted more stable than those of the silicon terminated (ST)-SiC/Cu from the results of the W ad and interfacial energies. The interfacial electron properties of SiC/Cu were studied via the charge density distribution, charge density difference, electron localized functions and partial density of the state. The covalent C-Cu bonds were formed based on the results of the electron properties, which further explained the fact that the interfaces of the CT-SiC/Cu are stable than those of the ST-SiC/Cu. The interfacial mechanics of the SiC/Cu were investigated via the interfacial fracture toughness and ultimate tensile stress, and the results indicate that both CT- and ST-SiC/Cu interfaces are hard to fracture. The ultimate tensile stress of the CT-SiC/Cu is nearly 23GPa, which is smaller than those of the ST-SiC/Cu of 25 GPa. The strains corresponding to their ultimate tensile stresses of the CT- and ST-SiC/Cu are about 0.28 and 0.26, respectively. The higher strains of CT-SiC/Cu indicate their stronger plastic properties on the interfaces of the composites.


Introduction
The copper metal composites have been extensively utilized in electronic technology [1], transportation [2]and aerospace fields [3]. Although the copper metal composites own high electrical conductivity, the weaker hardness and strength still limit their further applications. Therefore, copper alloy and copper matrix composites(CMCs) have been designed to improve their hardness and strength properties. However, as the reinforcement phase was introduced into the Cu matrix, the hardness and strength of the cooper materials are obviously enhanced [4,5]. Moreover, the reinforcements are the key factor of enhancing the mechanical properties for the CMCs without serious loss of the thermal and electrical properties of the matrix. Up to date, the reinforced phases including many types of ceramic materials, such as the carbide (TiC and WC) [6,7], the oxide (Al2O3, Y2O3) [8,9], and the ceramic (Ti3AlC2,AlN) [10,11] have already been employed to enhance the hardness and strength of the CMCs. Besides the ceramics discussed above, the iron [12] and its derivatives steel [13,14] were applied as the reinforcements for the copper matrix, which 3 of 20 also studied to figure out the interactions between the SiC partical phase and copper matrix.

Computational Details
In order to construct the 6H-SiC/Cu heterogeneous interfacial models, (0001) and (111) crystal surfaces were utilized to build the SiC, Cu slab and SiC/Cu composites models, respectively. The convergence of the Cu and SiC slabs were tested for acquiring the suitable atomic layers to meet the interiors features of SiC and Cu bulks. Moreover, along the c axis direction for SiC and Cu surface slab a 15 Å vacuum layer were added to eliminate the interactions between the surface atoms. In addition, for the sake of considering all possibilities of the SiC/Cu interfaces, the different stacking ways of Cu, the interior structures of SiC and different atomic terminations (carbon terminated (CT) and silicon terminated (ST) ) at the surfaces needing to be involved. Therefore, there are totally 18 types of SiC/Cu models displayed in Fig. 1. The stacking ways of the Cu atoms in Cu slabs along with the SiC surface, i.e., "HCP", "MT" and "OT" stacking ways were involved, in which "HCP" stacking way means: the interfacial Cu atoms placed on-top the first layer of SiC atoms; "MT" stacking way means: the interfacial Cu atoms which are reside atop of the connection midpoint of the first layer SiC atoms; "OT" stacking way means: the interfacial Cu atoms which are reside atop the second layer SiC atoms. Moreover, owing to the interior structure of the SiC, there are three different structurers connected with the interfacial Cu slab atoms, viz. "(a)" "(b)" and "(c)". The "(a)" of the 6H-SiC refers to the interfacial atoms C or Si (terminated) reside atop the midpoint of the atomic connection of the first layer of Cu atoms, "(b)" is invers of the structure "(a)", and "(c)" means that the interfacial C or Si atoms are connected on-top of the first layer Cu atoms. In general, three different Cu stacking sequences (HCP, OT and MT), three combined ways of SiC ((a), (b) and (c)) and two terminated Si and C atoms of the SiC slab at the surfaces were taken into account.
All calculations were based on the periodic boundary conditions and plane wave basis set and which carried out by the Cambridge Serial Total Energy Package Code (CASTEP) [41,42]. The Perdew-Burker-Enzerhof (PBE) functional generalized gradient approximation (GGA) [43] were performed to manage the exchange-correlation interactions. Moreover, the Monkhorst-Pack k-point grid [44] 11111 was sampled with the Brillouin zone for the SiC/Cu, Cu and SiC slab, respectively. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) [45] algorithm was applied to relax the atomic structures to reach the ground state. The 500 eV was chosen as the expansion in reciprocal space for cut-off energy of the plane wave. Nevertheless, the total energy tolerance, maximum force tolerance and maximal displacement such calculating convergent parameters were performed by 1.0 × 10 -5 eV/atom, 0.03 eV/Å and 1.0 × 10 -5 Å, respectively. The valence electrons of 3s 2 3p 2 , 2s 2 2p 2 and 3s 2 3p 6 3d 2 4s 2 were considered for the Si, C and Cu atom pseudopotentials. In addition, for all 6H-SiC/Cu interfacial models, 18 atoms were included during the whole processing of calculations.

Bulk properties
The fcc-Cu and 6H-SiC cells were optimized via GGA-PBE method [43] to evaluate the proper parameters, our calculated and the reported cell parameters listed in Table 1. The 6H-SiC belongs to the P63mc space group and its cell parameters are a = b = 3.095 Å, c=15.185 Å, α = β = 90, γ = 120, respectively. After optimization of the 6H-SiC cell, our calculated 6H-SiC cell parameters are a = b = 3.085 Å, c = 15.123 Å, α = β = 90, γ = 120, closed to the reported values in Table 1. The layer thickness was initially determined by testing the change of the layer distances, until the optimization of SiC (0001) and Cu (111) slab reached the convergence at the proper atomic layers. On the one hand, the results of the calculation might be inaccurate if atomic layers were too less. On the other hand, much time and resources would be spent if atomic layers were too large. Therefore, in order to acquire the approximate properties of the bulk for slab interior, the proper atomic layers of the slabs need to be pretested initially. The Δij is applied to determine the layer thickness of the slab and which can be  Table 2, in which the CT and the ST of the 6H-SiC slabs achieved the convergence with the tiny ij  values respecting to above 11 (for CT 34  is -0.97% , for ST 34  is -0.18%) SiC and 7( 23  is 0.72%) Cu layer thickness.
In specific, the calculated ij  values for all ST-6H-SiC layer thickness (most ij  are lower than absolute value |0.5%| ) are much lower than those of the CT-6H-SiC layer thickness, which show that the former are more inclined to be convergent than those of the latter. In addition, it can Owing to the hard directive detection of the 6H-SiC/Cu interfacial structure of the experiments, the simulative calculation is a useful method to analyze the interfacial structures details of the composites. The Si and C chemical potentials need to be considered when the 6H-SiC (0001) slab surface energies studied, because the polarization of SiC surfaces slab caused by these two types atoms. As a result, 6H-SiC (0001) surface plane can be divided into two components viz., ST and CT-surface. The surface energy ( s  ) can be defined in subsequent equation: where slab E refers to the total energy of the relaxed surface structure, slab Because the chemical potential of each element in the bulk is lower than that in the slab, the difference of the chemical potential for each element (Δμ) can be expressed via the following inequalities: The ST and CT-6H-SiC(0001) surface energies were calculated over the entire range of Si chemical potentials. Compared with the previous studies, our calculated 6H-SiC(0001) surface energies are respective as 2.553-2.878 Jm -2 (ST) and 7.367-7.692 Jm -2 (CT), which are a bit lower than the reported values 3.22 Jm -2 (ST) and 7.82 Jm -2 (CT) 31 . In addition, the reported 4H-SiC(0001) surface values are 2.899-3.535 Jm -2 for Si-terminated and 7.783-8.426 Jm -2 [55] for C-terminated. (other reported results are (2.86-3.52 for Siterminated and 7.92-8.59 for C-terminated)) [56] which are very close to our calculated results.

The surface energy of the Cu (111)
The surface energy of Cu (111) have been studied in many previous work, and their values are displayed in Table 3. Comparing with these surface energies, the Cu (111) surface energy is 1.39 J·m -2 calculated in this work, which closing to the reported values 1.32 J·m -2 [29],1.36 J·m -2 [30]and 1.40 J·m -2 [57], but higher than study results over the two decades (1.2 J·m -2 and 2.07 J·m -2 ) [58,29].

Work of adhesion (Wad) and interfacial distance
The work of adhesion (Wad) is utilized to evaluate the stabilities of the heterogeneous interfaces, which can be expressed as the work of separation when heterogeneous interfaces divided into the two free independent parts. Therefore, the Wad can be defined in the subsequent equation: In Eq. 10, where  In addition, interfacial distances of CT-6H-SiC(0001)/Cu(111) are larger than those of the ST-6H-SiC(0001)/Cu(111) in Fig. 3 (a) (b), and the average distance of the former is only 0.85 Å contrasting to the large average distance 2.0 Å of the latter. In specific, the most unstable interface CT-HCP(c) (0.4 J·m -2 ) own the largest interfacial distance 1.81 Å, and similar ST-OT(b) has the largest interfacial distance (2.34 Å) corresponding to its lowest Wad value (-1.81 J·m -2 ).

6H-SiC(0001)/Cu(111) interfacial energies
The stabilities of the heterogeneous interfaces can be evaluated via the interfacial energy. However, the assessment format of the interfacial energy is different with the work of adhesion, namely, the larger of the interfacial energies, the weaker of the interface stabilities. The interfacial energy can be defined in the following equation for the 6H-Si(0001)/Cu(111) composite models: In Eq. (11), where int   As shown in Fig. 5 (a), (b), (c), abundant charges assembled between the interfacial C and Cu atoms, which indicate that the strong interactions happened on these interfaces. However, in Fig. 5 (c), (d), (e), few charges distributed among the interfacial Si and Cu atoms, which reveal that the weak interactions taken placed on these two atoms at the interfaces. Compared with CT-6H-SiC(0001)/Cu(111) and ST-6H-SiC(0001)/Cu(111) interfacial charge distribution, it is noted that the charges are inclined to accumulated on CT (6H-SiC(0001)/Cu(111) interfacial atoms than those of the ST (6H-SiC(0001)/Cu(111), which show that the interactions between the Cu and C atoms are stronger than those between Cu and Si atoms.

Charge density difference of interfacial atoms of the 6H-SiC(0001)/Cu(111) interfacial atoms.
The charge density difference is applied to evaluate the charge communications between two atoms, which can be defined in following Eq. (12): Where the ρtotal is the total charge density of the 6H-SiC(0001)/Cu(111) interface, ρCu(111) and ρSiC (0001) refer to the charge densities of isolated Cu(111) and SiC(0001) slab, respectively. In Fig. 6, the blue color regions around Si and C atoms of the 6H-SiC indicate that few charge communications are performed in these regions. Conversely, the intertwined yellow and red colors surrounding the Cu atoms reveal that the strong electron communications have taken places among the interior Cu atoms. The interfacial atom Cu, Si and C have obviously interacted by their charge communications, the color difference between the interfacial Cu and C atoms are more apparent than those of the interfacial Cu and Si atoms, implying that charge communication work between Cu and C atoms are stronger than those between Cu and Si atoms.

Electron localization function (ELF) of 6H-SiC(0001)/Cu(111) interfacial atoms.
The ELF is a dimensionless and its values are between the ragion 0 and 1, which evaluated the electrons localized or un-localized state between the two atoms. When the ELF equal to 1, which means that the electrons between two atoms are fully localized, if ELF= 0, which show that the electrons totally un-localized, if ELF is equal to the median value (0.5), which indicates that the atoms surrounded via the homogeneous electron gases [60]. In Fig. 7, the dark green color among the Cu atoms of the six models reveal that abundant free electrons exist in Cu interior. Nevertheless, for interior SiC of the six models, the red color between Si and C atoms showing that strong covalent bonds formed of the two atoms. In addition, the color between the interfacial Cu, C and Si atoms are quite different with those of their interior, showing that bonding state of the interfacial atoms are various with those of the interior atoms. The partial density of state (PDOS) is applied to figure out the feature of the bonding states and electronic structures of the interfacial atoms. In this part, six 6H-SiC(0001)/Cu(111) type interfacial models are utilized to analyze the interfacial atom bonding states and their PDOS results displayed in Fig. 8. However, the PDOS of the other models can be seen in Fig. SI (7) to Fig. SI (8). In Fig. 8. (a), (b) and (c), the PODS of the fist C layer atom are different with those of their interior C layer atoms, for instance, a sharp peak appeared at -10.9 eV for s-orbital electrons of the 1 st C atoms in CT-HCP(a) and CT-MT(a) imply that the electrons belong to s-orbit of the 1 st C atom are larger than those of the interior C atoms. Moreover, similar intensity electron of the s-orbit for CT-OT(a) appeared a sharp peak at -10.1 eV contrast to CT-HCP(a) and CT-MT(a).
Nevertheless, the p-orbital charges of the 1 st C atom pass through the Fermi energy level, which show that the electrons un-localized between the interfacial C and Cu atoms. In comparison with the PDOS of interfacial and interior Cu atoms, the two peaks appeared at -1.51 eV and -1.99 eV for interior Cu atoms but which displayed at -3.74 eV and -4.69 eV for the 1 st Cu atom for CT-HCP(a) (Fig. 8(a)). The transformed differences of these two peaks are -2.7 eV and -2.23 eV, which show that the energies decreased more obvious than those of the interior Cu atoms. In addition, owing to the hybrid of the d-orbital electrons of the 1 st Cu atom with the p-orbital electrons of the  (Fig. 8. (a)), although the porbital electrons pass through the Fermi energy level, only feeble intension of the p-orbital electrons around the Fermi energy level. Contrasting to the s-orbital and p-orbital electrons of the interior Si atoms, subtle changes happened to those of the interfacial Si atoms, which imply that the p-orbital electrons of the interfacial Si atoms are less influenced by d-orbital electrons of the interfacial Cu atoms. Moreover, the line shape of the 1 st Cu dorbit is neither same as the those of the CT-HCP(a), nor as those of its interior Cu atoms. In comparison with the interior Cu d-orbit, the line shape of the interfacial Cu d-orbit has wider broad peak, which formed by two peak moved to each other. Therefore, the interactions between interfacial Si and Cu atoms are quite different with those of the C and Cu atoms, that is to say, the type bond between Si and Cu atoms are different with the bond which formed between Cu and C atoms. Moreover, the type bond formed between interfacial Si Fig. SI(7) to SI (8).

The interfacial elastic properties of 6H-SiC(0001)/Cu(111) interfaces
The elastic energy of the homogenous substance is a constant, while for heterogeneous materials are quite different [61][62][63]. For 6H-SiC(0001)/Cu(111) heterogeneous interfaces, the fracture toughness may tend to occur at the interface. Therefore, the interfacial fracture toughness were evaluated via the elastic energy and which defined in Eq.(12) [64]. In Eq. (12), where G1 and G2 refer to the shear modulus of SiC and fcc-Cu bulk, D1 and D2 represent the diameters of the atoms at the interface and ν2 is the Poisson's ratio of bulk fcc-Cu. The specific interfacial elastic values of 6H-SiC(0001)/Cu(111) are displayed in Table 5, and it can be found that the interfacial elastic energies of the CT-6H-SiC(0001)/Cu(111) are higher than those of corresponding ST-6H-SiC(0001)/Cu(111). According to the Table 5, the results further testify that the interfacial stabilities of CT-6H-SiC(0001)/Cu(111) are more stable than those of the ST-6H-SiC(0001)/Cu(111).
The assumed interfacial elastic energies are not consistent well with the sequence of the Wad and γ (interfacial energies), e.g., ST-MT(c) has the largest Wad (-1.32 J·m -2 ) and lowest interfacial energy (0.273-0.600 J·m -2 ), but its interfacial elastic energies (2.84 J·m -2 ) neither the highest nor the lowest among the ST-6H-SiC(0001)/Cu(111) interfaces, which mainly due to the complicated circumstance of the interfaces, such as diameter of the interfacial atoms, the occupation of the interfacial atoms and the work of adhesion of the interfaces. Similar results acquired for CT-6H-SiC(0001)/Cu(111) interfaces. Therefore, it can be noted that the elastic energies of 6H-SiC(0001)/Cu(111) interfaces influenced by different atomic termination (CT or ST) are stronger than those of stacking ways (HCP, MT, OT for Cu stacking and (a),(b),(c) for SiC interior structures). Specifically, the elastic energies of CT-6H-SiC(0001)/Cu(111) interfaces are about 2 J·m -2 larger than those of the ST-6H-SiC(0001)/Cu(111) interfaces. In addition, comparing with the same atom terminated interfaces (CT or ST-6H-SiC(0001)/Cu(111)), the difference of the elastic energies are no more than 1 J·m -2 , e.g., the difference of the highest (CT-OT(a)) and the smallest (CT-OT(c) elastic energy of CT-6H-SiC(0001)/Cu(111) is 0.87 J·m -2 (the difference of ST-6H-SiC(0001)/Cu(111) is 0.9 J·m -2 ).

Interfacial fracture toughness of the 6H-SiC(0001)/Cu(111)
The generation of the stress transferred depending on the interfaces, ascribed to the ductile matrix to brittle reinforcement of the composites. The energy released from the crack tip zone same to the energy required to form the crack area is a necessary condition for brittle fracture under the static condition.   Table 6. displays works of fracture values of all SiC/Cu interfacial models, and their values were obtained by using of the Eq. (14). According to the results in the Table 6., it can be noted that all int As strain increasing along the c directions of the models, the ultimate tensile stress of the various 6H-SiC(0001)/Cu(111) models could be carried out under the different ultimate strains. Therefore, to make clear of the ultimate tensile strength of the 6H-SiC(0001)/Cu(111) interfacial models, a 0.02 strain step performing on c directions until to reach the ultimate tensile stress of the 6H-SiC(0001)/Cu(111) interfaces. In order to obtain the relationships of the strain and ultimate tensile stress, the stress vs. strain were plotted to acquire the variation trend of the stress. In this work, the normal strain can be expressed by Eq. (16) [64] in terms of engineering strain.
In Eq. (16), Where l0 and l refer to the primary cell length and the deformed cell length, respectively. The engineering strain yield and keep onto the interfacial supercell model in a quasi-static way. The ultimate tensile stress of the C-terminated and Si-terminated 6H-SiC(0001)/Cu(111) can be obtained via the plot of the stress vs. strain in Fig. 9.
The Fig. 9 (a) (c) contain all strains which are various from 0 to 0.32, and over the highest stress a 0.005 strains step was added to confirm the ultimate tensile stress. The Fig. 9 (b) and (c) are enlarge graphs which are marked red square in Fig.9 (a) and (c) respectively.
According to the Fig. 9, the plotted strain vs. stress of CT-6H-SiC(0001)/Cu(111) can be distinguished by their color point lines ( Fig. 9 (a). However, the plotted strain vs stress of ST-6H-SiC(0001)/Cu(111) have the same variation trend which lead to all color point lines overlapped and which cannot be distinguished (Fig. 9 (c). The ultimate tensile stress for CT-6H-SiC(0001)/Cu(111) interfaces are nearly at 23 GPa (which are various from 22.11 GPa (CT-OT(b) to 23. 73 GPa (CT-HCP(c)) and their corresponding stains are different from 0.26 (CT-HCP(c)) to 0.295 (CT-HCP(b)). In addition, the ultimate tensile stress for ST-6H-SiC(0001)/Cu(111) in Fig. 9 (d)  corresponded to the ultimate tensile stress with the small differences for ST-6H-SiC(0001)/Cu(111) interfaces, and they are respective from 0.26 to 0.27. In comparison with the ultimate tensile stress of the ST-6H-SiC(0001)/Cu(111), the CT-6H-SiC(0001)/Cu(111) own higher ultimate tensile stress values. The strains at the ultimate tensile stress are equal to or higher than 0.28 (excepting for CT-HCP(c) (0.265)) for most of the CT-6H-SiC(0001)/Cu(111) interfaces. The strain corresponded to the ultimate tensile stress of CT-6H-SiC(0001)/Cu(111) interfaces are larger than those of the highest strain corresponded to the ultimate tensile stress 0.27 of the ST-6H-SiC(0001)/Cu(111) interfaces. The higher strain of the CT-6H-SiC(0001)/Cu(111) interfaces indicate that they have better plastic properties, which ascribe to the C-Cu formed at the interfaces. Namely, the plastic properties have been enhanced by SiC reinforcement of the copper matrix, which are mainly due to the covalent carbide are formed at the interfaces.