*Using first-principles calculations, the electronic structures and optical properties that arise on doping-atom-containing silicon nitride systems are reported as a function of dielectric constant, reflectivity, absorption and loss spectra.
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**BY XUEFENG LU, XIN GUO, PEIQING LA, YUPENG WEI, JIANBO YIN and XUELI NAN, Lanzhou University of Technology, Lanzhou, China**

The development of hexagonal silicon nitride is concerned predominantly with the design and fabrication of structural materials because of its excellent properties such as high strength, toughness, flexibility and good resistance to thermal shock and oxidation [1-3]. While a great deal is understood of block Si_{3}N_{4} behavior, one of the challenges to exploiting the functionality will be to better understand the variety and corresponding structural features unique to Si_{3}N_{4}. Eliciting a desired output from controlled inputs using predesigned materials is one of the key aspects of current materials’ research field.

A first-principles calculation within the local density approximation suggested that -βSi_{3}N_{4} has a direct band gap of about 4.34 eV. Although the calculated band gap may be somewhat underestimated, owing to the LDA, β-Si_{3}N_{4} is likely to fall into the category of wide-gap semiconductors by appropriate doping [4-5]. Simultaneously, the direct band gap reveals the potential applications as optical or electric devices apart from the structural applications. In order to use β-Si_{3}N_{4} as a semiconductor, it is necessary to find proper dopants.

Recently, there was substantial interest in investigating the properties of Si_{3}N_{4} that combined the adsorption and doping. Various theoretical methods have been employed to model and understand these behaviors, motivated by the fact that different structures may exhibit desired and fascinating properties. Wang simulated the interaction of O2 with the β-Si_{3}N_{4} surface and found that the oxidation occurs on the-SiN (0001) surface at 1200°C more easily 34 by means of the calculated significant chemisorption energy and the short adsorption bond length [6]. Oba investigated n- and p-type dopants for cubic silicon nitride and concluded that P and O are preferable for n-type doping, while Al is favorable for p-type doping in terms of the formation and ionization energies [7]. The dependence of the formation energies on the chemical potentials indicates that a proper choice of growth conditions is mandatory for suppressing the incorporation of these impurities into anti and interstitial sites [8]. For the low Al concentration, the material exhibits the dielectric behavior, while the metallic behavior for the high Al concentration [9]. Ching concluded that cations with small radii tend to occupy the tetrahedral site and those with large radii tend to occupy octahedral sites for γ-Si_{3}N_{4}, which is likely to show some metallic characteristics at the Ti concentration [10].

Previously, we have reported the impurity species appropriate to β-Si_{3}N_{4} based on the first-principles calculations [11-13]. In the present contribution, we conducted similar calculations to investigate impurities effects in β-Si_{3}N_{4}, as well as the electronic structures and optical properties of rare earth (RE)-doped systems, to obtain the further information in details.

**Theoretical approaches and computational procedures**

Here, we demonstrated the first-principles calculations using the CASTEP program code with plane-wave pseudopotential (PWP). Our simulation model, detailed in the method section, is summarized in **FIGURE 1** and described briefly here. The hexagonal β-Si3N4 unit cell contains two formula units (14 atoms) with a space group P63/m. The idealized Si-N layers in an ABAB sequence can depict the structure perfectly. All of the Si atoms are equivalent (6h sites), but there are two inequivalent nitrogen sites: N2c at 2c sites and N6h at N6h sites. The N2c atoms locate in a planar geometry with their three Si nearest neighbors, and N6h atoms locate at slightly puckered sites enclosed by three Si atoms, while Si atoms locate at the center of slightly irregular tetrahedron bonded with one N2c atom and three N6h atoms. Supercells (2×1×1) containing 28 atoms were used to simulate RE-doped β-Si3N4. The doping was realized by substituting some Si atoms with Gd and Tb atoms. The core region and valence electrons of the atoms in the systems were illustrated by the ultrasoft pseudopotential. The lattice constants of primitive cells were determined through calculations using a plane-wave cutoff energy of 770 eV and a 4×4×10 point grid in the irreducible part of the Brillouinzone. The structural optimization was done by relaxing both internal coordinates and the lattice constants by calculating the ab initio forces on the ions, within the Born-Oppenheimer approximation, until the absolute values of the forces were converged to less than 10-2 eV/Å. The stress level for the equilibrium structure is less than 1×10-2 GPa and the max displacement is 5×10-4 Å in supercells.

**Results and discussion**

**Geometry optimization results:** Both models are composed of 28-atom, for which the only difference is rare element. The optimization was carried out by relaxing both the internal coordinates and the lattice constants by calculating the ab initio forces on the ions, until the absolute values of the forces were smaller than the set values. The obtained results are summarized in **Table 1**. Compared to the experiment values, the calculated lattice constants are overestimated within 1%, illustrating a reasonable agreement. Because of the bigger atoms radius of Gd and Tb atoms, the lattice constants and volume increase remarkably after doping. The Eg of Gd-doped system is 0.095 eV, and decreases to 0.073 eV for Tb-doped one due to the narrower distance of the bottom of conduction band and the top of valence band derived from doping. For the purpose of characterizing the stability of doped structures, the Eb and Ef are brought forward in terms of total energy and total sums of elements free energies, and the atomic chemical potential, respectively, as illustrated in Eqs. (1) and (2).

where ESi_{3}N_{4} and ET denote the total energies of the supercells before and after doping, respectively. X is doping atom, μSi and μX are the chemical potentials of Si and doping atoms, respectively. Note that Eb of system accounts for its stable degree. The and the conduction band is about 0.1 eV, which is adequately in reasonable accordance with the results of energy band structures as discussed above. For Tb-doped system, also three valence band parts can be observed: the bottom band ranging from -43.03 to -41.73 eV originates mostly from Tb s orbital electrons; the next one in the middle of -21.71 and -12.31 eV is briefly due to Tb p and N s states; the larger the absolute value is, the more stable the final structure is. The stability of Gd-doped system with small Eb is higher than that of Tb-doped one. The Ef is relative to the chemical potentials of elements and may also exhibit the stability of doped structures. The obtained Ef values for Gd- and Tb-doped systems are 12 and 15eV, respectively, revealing that the structure of the former is more stable than that of the latter.

**Electronic structures: FIGURES 2A** and **B** show the achieved electronic band structures of Gd- and Tb-doped supercells. For Gd-doped system, the Eg drops obviously to 0.095 as compared to undoped one (4.336 eV). The valence band (VB) can be divided into three group: the bottom of VB is about -40.94 eV; the next one is in the range of -20.9 and -12.8 with a width of 8.1 eV; the top VB locates between -9.62 and 1.15 eV with an extent of 10.77 eV. While for the Tb-doped system, the Eg decreases continu- ously to 0.073. The VB may also be divided into three group: the bottom of VB is at -42.45 eV; the next one locates between -21.29 and -12.79 eV accompanying a width of 8.5 eV; the top VB is in the range of -9.75 and 0.89 eV with a width of 10.64 eV. It is worth noting that both VBs have higher densities than those of undoped system accompanying the overlap of the energy band although doping with a lower concentration.

In order to analyze further the results according to the band structures, we conduct the densities of states (DOS) coming from the calculations with GGA, as demonstrated in **FIGURE 3**. The calculated total DOS derives from the partial density (PDOS) of N, Si, Gd, and Tb atoms. The obtained total and partial densities of states for Gd- and Tb-doped systems are displayed in Fig.3a and b, respectively. One can see that for Gd-doped system, there are three valence regions: the lower energy band located between -41.46 and -40.06 eV briefly comes from Gd s orbital electrons; the next energy band in the range of -21.34 and -11.95 eV mainly originates from Gd p and N s orbital electrons; the upper valence band occurs between -9.99 and 1.74 eV is primarily from Gd f and N p states; the conduction band in the range of 3.23 to 6.40 eV principally consists of Si p orbital electrons. The distance between the top valence band top valence band located between -10.07 and 1.38 mainly is owing to Tb f and N p orbital electrons; the conduction band between 2.85 and 6.68 eV is composed of Si p states. Summing up, the above, RE doping contributes to the formation of the lower VB of the doped systems, derived from the s orbital electrons of RE, and the formation of narrow band gap, revealing that potential applications in semiconductor devices.

To explore the insights into the comprehension of the charge transfer of both systems, the electron density difference maps in planes containing different atoms are displayed in **FIGURES 4** and **5**. The emerged blue and red parts revealed in pictures represent the electron loss and gain, respectively. It can be seen that the changes in electron density are apparent when Si atom is substituted by Gd or Tb atoms, especially for Tb doping. In the case of both cases, the electron loss, which occurred near the N atom between Si and N atoms and is close to doped atoms, weakens after doping, while the electron loss turns into electron gain with regard to Tb intervention compared to undoped field, weakening the strength of the covalent bond. As we investigate more carefully these maps, the charge density distributions of non-spherical in both cases can be observed. Concerning the electron loss, the toothed shapes that the pentagonal starfish and latin cross can be seen for the Gd- and Tb-doped systems, respectively.

**Optical properties investigation:** The optical properties of doped systems are not perfectly understood at all owing to the two particularly challenging problems. Experimentally obtaining the single-crystal samples and property response is difficult and theoretically the works of optical properties of element doped silicon nitride are lack. Due to optical properties not only contain the occupied and unoccupied parts of the electronic structures but also carry the information about the character of bands, it is of underlying importance. In this work, we carry out a complete analysis of doped systems based on first-principles spectroscopy for different optical functions such as imaginary and real parts of the dielectric function, the reflectivity, the absorption spectra and the loss function. Due to the considered systems crystallize in the hexagonal structure with space group P63/m, the dielectric tensor contains three components corresponding to the electric field along the a, b, and c-crystallographic axes, i.e. εxx, εyy, and εzz. The imaginary part ε2(ω) is given in equation (3), and the real part ε1(ω) can be derived from the imaginary part employing the Kramer- Kronig transformation as shown in equation (4). The absorption coefficient (ω) and the electron energy loss function L(ω) can be gained directly related to ε1(ω) and ε2(ω) as described in equation (5) and (6).

**FIGURE 6** shows the real ε1(ω) and imaginary ε2(ω) parts of theoretical dielectric function of supercells doped by Gd- and Tb-doped systems. It is meaningful parameter due to the reason that it embodies the basic feature of linear response to an electromagnetic wave and determines the only propagation behavior of the radiation within. One can see that the static dielectric constant obtained at the zero frequency of the real part decreases to 7.97 after Gd doping, and markedly increases to 10.5 for Tb doping with respect to undoped system (8.2) [16], revealing its potential applications in electrics and optics. Compared to the other two directions, εyy and εzz has larger values of 10.76 and 16.1 for Gd- and Tb-doped systems, respectively. Correspondingly, the change is similar to that of imaging part. This reveals that Gd-doped system can exhibit longer life in application as dielectric materials in the low energy regions because of the low static dielectric constant and loss.

**FIGURE 7** illustrates the absorption spectra η(ω) of doped systems. It is found that the strong absorption edges locate between 5 and 16 eV, giving the threshold for direct optical transitions between the top valence band and bottom of conduction band. All the parts Ai (i=t, xx, yy and zz) display main peaks located at 10.52, 10.47, 10.51 and 9.19 eV for Gd-doped system and 10.55, 10.71, 10.71 and 10.49 eV for Tb-doped system. On the left of the host absorption peaks, the other peaks (about 1.78 and 25eV) appear, which are attributed to the interband transitions of free electronic carries in the top of VB. The values of the peaks are lower and have a small effect on the host peak, indicating that in the low energy region the doping systems may still exhibit “Transparent Type” compared to undoped system [13], but the range of the edges of the absorption peaks reduced. The obtained reflectivity spectra detailed in **FIGURE 8** is displayed. Spectra profiles are similar for Gd- and Tb-doped of equal peak situations. At the same time, two host peaks all locate at 11.7 eV. It is worth noting that three components Ri (i=xx, yy and zz) display the similar peaks value and positions, which reveals that the optical properties studied here show the characteristics of some isotropy.

**FIGURE 9** summarizes the theoretical electron energy loss spectra (EELS) L(ω) of doped systems, which is in agreement with the imaginary part of the reciprocal of the dielectric function. The peak of EELS is related to the plasma resonance and the frequency interrelated with the peak is the so-called plasma frequency, above which the material exhibits the dielectric behavior, while below which the material behaves like semiconductors and metals.

It is found from the results that the host peaks of doped systems are at about 12.5 and 14 eV, respectively, which are lower than that of undoped system (20 eV), indicating that a red-shift phenomenon is present after doping. At the same time, one weak peak appears at 2 eV. As can be seen from the calculation results that the doped systems may display the semiconductor behavior, which is in agreement with our calculated values of the static dielectric constants. At the same time, it is also concluded that light spreads easily in the lower energy regions for the doping systems.

**Conclusions**

In summary, using the first-principles calculations, we report on the electronic structures and optical properties that arise on doping-atom-containing silicon nitride systems as a function of dielectric constant, reflectivity, absorption and loss spectra. The results are as follows:

(1) The fully relaxed structural parameters are found to be in good agreement with experi- mental data. Calculated banding energies of Gd-and Tb-doped systems are -204 and -197 eV, and formation energies are 12 and 15 eV, respectively. It can be readily seen that the structure of the former is more stable than that of the latter.

(2) The electron loss near the N atom between Si and N atoms turns into electron gain with respect to Tb intervention compared to undoped field, weakening the strength of the covalent bond. Concerning the electron loss, the toothed shapes that the pentagonal starfish and latin cross can be observed for the Gd- and Tb-doped systems, respectively.

(3) The absorption band ranges of doped systems become narrower. Both of reflectivity spectra profiles are similar and all locate at 11.7 eV, exhibiting the characteristics of some isotropy. In theoretical electron energy loss spectra, the host peaks of doped systems locate at about 12.5 and 14 eV, indicating that a red-shift phenomenon occurs after doping.

(4) Gd-doped system can exhibit longer life in application as dielectric materials in the low energy regions because of the low static dielectric constant and loss, as well as transparent type characteristic exhibited in lower energy region.

**Acknowledgements**

The work was supported by The National Natural Science Foundation of China (51402142, 51164022), The Gansu Provincial Youth Science and Technology Fund Projects (1310RJYA006, 1212RJYA004), Gansu Provincial Science and Technology Support Program (1304GKCA027), Gansu Provincial Department of Construction Project (1201ZTC042).

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*XUEFENG LU, XIN GUO, PEIQING LA, YUPENG WEI, JIANBO YIN and XUELI NAN are researchers at the State Key Laboratory of Gansu Advanced Non-ferrous Metal Materials, Lanzhou University of Technology, Lanzhou, China. JIANBO YIN is also with the State Key of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Science, Lanzhou, China.*