Introduction

BN nanotubes (BNNTs) were theoretically predicted in 1994 [1] and then were experimentally realized in the following year. [2] The honeycomb arrangement of BNNT is similar to carbon nanotube (CNT), but B and N atoms completely substitute for C atoms. Contrary to CNTs that can be a semiconductor or a metallic nanostructured materials, BNNTs are electrical insulator materials with a constant band gap around 5.5 eV and almost independent of their radius and tube chirality. [3,4,5] This property has been primarily interested in studying the properties of BNNTs such as tuning the electronic properties. The chemical and physical methods can be utilized for tuning the electronic properties of BN nanotubes [6]. The physical methods include applying electric field [7] and strain [8], and the chemical methods contain doping, [9,10,11,12,13,14,15,16,17,18,19,20] introducing defect [21], functionalization [22, 23], absorption [24] and so on.

Kim et al. [25] have showed that the formation energy of a 2C-doped SW-BNNT is approximately 3.1 eV lower than that of the un-doped SW-BNNT having B–N bond. The NO adsorptions on the surface of C-doped SW-BNNTs are energetically favorable and are stronger than pristine and un-doped SW-BNNTs [24]. Due to the introduction of Pt states into the nanotube band gap, Pt-doped BNNTs are more reactive than pristine and SW defective BNNTs [26]. It has been reported that the adsorption of HCOH molecule on both Si-substituted boron defect site and Si-substituted nitrogen defect site of the BNNT is chemisorption, while its adsorption is predicted to be weak physisorption on the pristine BNNT. Besides, the Si-doped BNNT shows the high sensitivity to toxic HCOH [27].

The Stone–Wales (SW) transformation is a topological defect. Upon formation of this type of defect in BNNTs, B–B and N–N bonds are created and the formation of these bonds increases the total energy of the NT [28]. However, atomistic simulations [29, 30] and experiments [31, 32] have shown that the SW defect leads to the formation of 5–7 ring pairs in the hexagonal lattice of BNNTs. In addition, SW defect can be created in a BNNT under axial tension [32, 33].

The easy modifying of electronic properties of BNNTs is still a challenging task. The elastic strain engineering (ESE) is one of the nonchemical tools for adjusting the electronic properties. The ESE allows us to control the band gap of materials by simply applying an elastic strain. Effects of axial tension on the electronic properties of SW defective BNNTs were previously investigated [34,35,36,37,38]. Gupta et al. [39] were explored the impact of Si doping on the electronic structure and electron transport properties of boron nitride monolayers. Wei. et al. [40] have experimentally investigated the influence of tensile loading and pullouts on the properties of individual multiwalled BNNTs for the first time. Deformation behaviors of an (8,8) BNNT under axial tensile strains were investigated via molecular dynamics (MD) simulations by Liao et al. [41]. They reported that the BNNT starts to fail at the failure strain of 26.7%. The effect of axial tensile strain on structural and electronic properties of zigzag BNNT with different length and diameter was studied [8]. Ge et al. [42] found that the band gap of (n = odd number, 0) B2C tubes decreases to zero by increasing tensile strain.

The study of the sensitivity of nanotubes toward different gases is at the center of deep experimental and theoretical researches [43, 44]. The Si-BNNTs were used as a metal-free catalyst for oxidation of NO [45]. Zhang et al. [46] fabricated a NO gas sensor employing multiwalled carbon nanotubes (MWCNTs). The adsorptions of CO and NO molecules on the surface of transition metals (V, Cr, Mn, Fe, Co or Ni)-doped (8,0) BNNTs were investigated by using first-principle calculations [47]. Moreover, SiC-based nanodevices were predicted to be useful for the design of the NO and CO sensors [48].

The effect of the Stone–Wales (SW) defect on the response of BNNT to axial tension was studied in our previous work [34]. In addition, we have investigated effect of C doping on the properties of SW-BNNTs [24]. In the first part of this work, the influences of simultaneously Si doping and axial tensile strain on electronic and structural properties of three Si N , Si B and Si N,B -doped SW defective BNNTs are explored. To the best of our knowledge, no experimental or theoretical investigation has been reported on the sensitivity of strained and unstrained Si-doped SW defective BNNTs toward the NO gas. In the second part of the present work, reactivity and sensitivity of strained and unstrained Si-doped SW defective BNNTs toward NO gas were investigated.

Computational details

The DFT method was used to study the effect of axial tensile strain on the electronic and structural properties of Si-doped SW defective BNNTs. Three models of (6,0) zigzag single-walled Si-doped SW–BNNT consisting of B35N36SiH12 (Si B ), B36N35SiH12 (Si N ) and B35N35Si2H12 (Si N,B ) that the Si atoms are doped in the region of SW defect were explored. The structure NTs are shown in Fig. 1. All geometry optimizations were performed using the M05-2X functional [49, 50] and 6–31+G(d) basis set as implemented in the computer program packages [51, 52]. In the first step, the structures were optimized without any restriction. Then, the axial tensile strain is applied on optimized Si-doped SW–BNNT. No symmetric constraints are imposed during the optimization. In the axial stretching process, distances between B and N atoms at both ends of Si-doped SW–BNNT were scanned by 0.1 Å increments at nineteen steps and remaining degrees of freedom were relaxed during the calculations.

Figure 1
figure 1

Optimized structures of a SW defective (6,0) BNNT, b B35N36SiH12 (Si B ), c B36N35SiH12 (Si N ) and d B35N35Si2H (Si N,B ). The B, N and Si atoms are represented in pink, blue and silver, respectively. The SW defects are highlighted in ball-stick representation

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The binding energies (BE) per atom for Si-doped SW defective BNNTs are calculated according to the following formula:

$$ {\text{BE}} = \frac{{\left[ {aE_{\text{B}} + bE_{\text{N}} + cE_{\text{H}} + dE_{\text{Si}} } \right] - \left[ {E_{{{\text{B}}_{a} {\text{N}}_{b} {\text{H}}_{c} {\text{Si}}_{d} }} } \right]}}{a + b + c + d} $$
(1)

where a, b, c and d are the number of B, N, H and Si atoms. E B, E N, E H, E Si and \( E_{{{\text{B}}_{a} {\text{N}}_{b} {\text{H}}_{c} {\text{Si}}_{d} }} \) e the ground state electronic energies of B, N, H, Si atoms and optimized doped BNNT, respectively. The engineering strain in the axial direction of the doped SW–BNNT is given [53] by

$$ \varepsilon_{\text{Eng}} = \frac{{L - L_{ 0} }}{{L_{0} }} $$
(2)

where L 0 is the initial length prior to loading and L is the nanotube length upon the strain.

The formation energies of the Si-doped SW defective BNNTs were estimated using Eq. (3) given below

$$ E_{\text{form}} = E_{\text{d}} + \left( {a + b} \right)E_{{{\text{B}}\,{\text{or}}\,{\text{N}}}} - mE_{\text{Si}} - E_{\text{p}} $$
(3)

where E d is the calculated total energy of BNNT containing defects, E B or N is the total energy of host boron or nitrogen atoms removed from the nanotube, E Si is the total energy calculated for the atomic Si, E P stands for the total energy calculated for the pristine BNNT and a, b and m are the number of B, N and Si atoms, respectively. The band gap is obtained from the difference between the orbital energies of the LUMO (conduction band minimum) and the HOMO (valence band maximum). To plot the density of states (DOS), we used the Multiwfn software [54].

Global reactivity descriptors measure the overall reactivity of a molecule. Some of descriptors are chemical potential, chemical hardness, global softness, etc. Chemical potential plays an especial important role in semiconductor physics [55]. The DFT-based reactivity descriptors are good prediction tools for studying reactivity especially in probing the regiochemistry of different types of chemical reactions [56,57,58].

The relation of chemical potential (µ) and the electronegativity (χ) [59, 60] can be written as follows:

$$ \mu = - \chi = - \frac{1}{2}\left( {I + A} \right) $$
(4)

The global chemical hardness (η) is defined [61] as:

$$ \eta = \frac{1}{2}\left( {I - A} \right) $$
(5)

where I and A are the first ionization energy and electron affinity, respectively. The chemical meaning of the word “hardness” is the resistance of the chemical potential to change in the number of electrons.

In the finite difference approximation, the ionization energy and electron affinity can be replaced by the E HOMO and E LUMO, respectively:

$$ \mu = - \chi = \frac{1}{2}\left( {E_{\text{HUMO}} + E_{\text{LUMO}} } \right) $$
(6)
$$ \eta = \frac{1}{2}\left( {E_{\text{LUMO}} - E_{\text{HOMO}} } \right). $$
(7)

The electrophilicity index (ω) [62], which measures the capacity of an electrophile to accept the maximal number of electrons in a neighboring reservoir of electron sea, is defined according to the following equation

$$ \omega = \frac{{\mu^{2} }}{2\eta. } $$
(8)

The chemical softness (S) is defined as following equation

$$ S = \frac{1}{\eta }. $$

NO adsorption on strained and unstrained Si-doped SW-BNNT is explored at M05-2X/6–31+G(d) level of theory. In addition, dispersion-corrected AEs at M05-2X-D3/6–31++G(d,p)//M05-2X/6–31+G(d) level of theory were also calculated in order to find the effect of dispersion and improved basis set on the interaction energy between NO- and Si-doped BNNT.

Results and discussions

Geometric and energetic descriptions of Si-doped (6,0) SW defective BNNTs

As shown in Fig. 1a, there are two unfavorable homonuclear B–B and N–N bonds in a SW defect region that are connected by B–N bond. In the Si B , Si N and Si N,B structures, B atom, N atom and both N and B atoms of this bond were replaced with a Si atom, respectively. As shown in Fig. 1, when the SW defect is created, three types of B–N bonds can be found: longitudinal B–N bonds; parallel to the longitudinal direction of the tube that are designed as L 1–L 6, zigzag (diagonal) B–N bonds that are labeled as L 7–L 18 and one B–N bond orthogonal to the tube axis that is designed as L 19 (bond connecting the two pentagons and two heptagons in 5–7–7–5 ring fusion).

The experimental studies on the Si-doped multiwalled BNNTs have revealed the formation of Si–B and Si–N bonds in the lattice [63, 64]. In the Si B - and Si N -doped SW–BNNTs, one B and N atoms of the vertical B–N bond in the SW defect region was substituted by one Si atom. After structural optimizations, we found that Si doping causes significantly large distortion in defect region. The bond length values of L 1–L 19 bonds and selected bond angles around the substituted atoms are listed in Table 1. The length of Si–N (L 16), Si–B (L 15) and Si–B (L19) sp 3-like bonds in Si N nanotube (NT) is 1.792, 2.023 and 1.975 Å and that of Si–N (L 17), Si–B (L 18) and Si–N (L 19) bonds in Si B nanotube is 1.779, 2.028 and 1.784 Å, respectively, which are much larger than the N–N (L 16; 1.449 Å), B–B (L 18; 1.713 Å) and B–N (L 19; 1.434 Å) sp 2 bonds in un-doped SW-BNNT. The vertical L 19 bond length is 1.434 Å, 1.975 and 1.784 Å in the SW–BNNT, Si N and Si B nanotubes, respectively, indicating that vertical B–N bond in un-doped nanotube is shorter than vertical Si–B and Si–N bonds in Si-doped ones. Therefore, substitution of Si atom increases L 19 bond in both models, so that this increase for Si N is greater than Si B . Since the atomic radius of Si atom is larger than N and B atoms, substitution of N and B by Si in the vertical L 19 bond increases its bond length. The two other bond lengths between Si and neighboring atoms (L 16 and L 15 in Si N and L 17 and L 18 in Si B ) are longer than those found in SW–BNNT.

Table 1 Selected bond lengths (Å) and bond angles calculated at M05-2X/6–31+G(d) level for Si-doped SW-BNNTs
figure a
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In Si N,B NT, both N and B atoms in L 19 bond are replaced with two Si atoms. The calculated L 15–L 19 bond lengths in Si N,B NT are 2.019, 1.751, 1.876, 2.138 and 2.245 Å, respectively. Since the size of Si atom is greater than B and N atoms, optimized L19 bond length at the 7–7 ring fusion of the Si N,B (2.245 Å) is greater than those of Si B (1.784 Å), Si N (1.975 Å) and un-doped NT (1.434 Å). Therefore, L 19 bond in Si B,N NT is weaker than that of Si N and Si B NTs. Two other bonds (L 15 and L 16 in Si N ) and (L 17 and L 18 in Si B ) are, respectively, longer and shorter than those of corresponding bonds in Si N,B NT.

In addition, bond angles around the Si atom in Si-doped NT change with respect to un-doped one. The sum of three angles around the N and B atoms involved in vertical B–N bond in un-doped nanotube is 341.4° (average bond angle of 113.8°) and 352.2° (average bond angles of 117.4°), respectively, indicating that the N atom is out of tube surface with respect to B atom. When that Si → (B or N) substitution happens, sum of the three angles around the Si atom in Si B and Si N decreases to 294.7° and 296.3°, respectively, which are smaller than sum of the normal pyramidal angles. Therefore, Si atom in Si N and Si B NTs preserves pyramidal-like electronic geometry sp 3 with the average bond angles of 98.2° and 98.8°. The increase in bond lengths and a decrease in bond angles around the Si atom forces the Si atom to protrude outwardly from the tube surface (see front view of Fig. 1).

Comparison of the bond angles in Si N,B NT shows that the sum of three bond angles around the Si atom of the Si → N substitution (347.6°) is greater than that of Si → B substitution (247.4°). In other words, average bond angle around the Si atom of the Si → N substitution (115.9°) is greater than that of Si → B substitution (82.5°). Therefore, it is expected that the distance of Si atom in Si N,B NT from the surface of NT in Si → B substitution to be greater than that of Si → N substitution, as can be seen in Fig. 1. The average bond angle values around the Si atoms in Si N,B NT (115.9° and 82.5°) are greater/smaller than those of estimated for Si N (98.2°)/Si B (98.8°). The calculated E angle of Si B NT (105.3°) that almost is consistent with the previous reports [41, 65].

In pristine BNNTs, N atoms and their nearest three B atoms form local pyramid structures. They are not located on the same cylindrical surface so that outer and inner shells occupy by the N and B atoms, respectively [7, 32, 66]. Thus, the radial geometry of the tubular structure is characterized by two concentric cylindrical tubes: all of B atoms forming the inner cylinders and all of the N atoms forming the outer cylinders. The radial buckling (β) is defined [41] by

$$ \beta = r_{\text{N}} - r_{\text{B}} $$

where r B and r N represent the radii of the B and N cylinders. If the value of β approaches zero, the B and N atoms will be located on the cylindrical surface of the BNNT, while a positive value indicates that the BNNT consists of two cylindrical surfaces with N atoms situated on the outer surface. In (6,0) SW-defected BNNT and Si-doped SW-defected BNNTs, we have defined three types of β at the different areas of Si-doped SW defective BNNTs as shown in Fig. 2. The average value of β for Si B , Si N and Si N,B is 0.096, 0.088 and 0.084 Å, respectively, indicating that difference in radii of the outer and inner cylinders in Si B is greater than others and in Si N,B is the smallest value.

Figure 2
figure 2

Three defined radial buckling (β) in NTs

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The calculated BE and defect formation energies (E form) for SW and Si-doped SW defective BNNTs are reported in Table 2. As can be seen, the BE values for Si-doped SW defective BNNTs Si B (7.11 eV), Si N (7.08 eV) and Si N,B (7.07 eV) NTs are smaller than that of BE of SW–BNNT (7.14 eV).

Table 2 Average radial buckling (β), binding energy (BE), defect formation energy (E form), HOMO (E H), LUMO (E L) and band gap (E L–H) energies for NTs calculated using M05-2X method
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The calculated defect formation energies (E form) of Si B , Si N and Si N,B NTs are 4.96, 6.95 and 7.59 eV, respectively, indicating a greater probability for the Si atom to replace the boron atom than the nitrogen atom. As can be seen, the formation energy for Si N,B NT is greater than Si B and Si N ones. Therefore, substitution of vertical B–N bond in the SW defect region by Si–Si bond leads to a nanostructure that is less stable than two others. There is a correlation between L 19 bond length in Si B , Si N and Si N,B NTs and their E form; defect formation energy increases from Si B to Si N and Si N,B NTs as the L 19 bond length increases.

The energies of the highest occupied molecular orbital (E HOMO), the lowest unoccupied molecular orbital (E LUMO) and the energy gap between LUMO and HOMO are tabulated in Table 2. The Si N,B NT has the lowest value of E HOMO and E LUMO compared to the other two NTs. The energy gap for Si N,B NT (4.85 eV) is smaller than Si N (5.43 eV) Si B (5.57 eV) and un-doped SW-BNNT (6.82 eV). Therefore, order of energy gap is SW-BNNT > Si B  > Si N  > Si N,B . The smallest value of the energy gap in Si N,B NT can be attributed to smaller value of LUMO energy in Si N,B (−2.11 eV) compared with others. From a comparison of energy gaps, it can be concluded that the conductivity of Si N,B NT is the greater and excitation energy of Si N,B NT is smaller than other two NTs.

Isodensity surfaces of HOMO and LUMO for SW–BNNT, Si B , Si N and Si N,B NTs are given in Fig. 3. As can be seen, electron density of the HOMO of SW-BNNT is localized on the B–N bonds at the defected area and slightly on the nitrogen atoms in the vicinity of the defected region, but the electron density of HOMO of Si B model mainly localized on the Si atom. Also electron density of LUMO of both SW-BNNT and B–Si model is distributed on the B–N pairs along the tube axis and mainly on the end of the B-terminal. Thus, it is expected that electrophilic sites of SW–BNNT and B–Si models are distributed on the B–N pairs at the end of B-terminated and their nucleophilic site is distributed on the Si atom in the center of defect region. The isodensity surface of the HOMO of Si B,N NT is almost similar to HOMO of Si B model and its distribution is concentrated on Si atom of the vertical bond in the SW defect region. Also, the isodensity surfaces of the LUMOs of Si N,B NT have similar behavior for the isodensity surface of the LUMOs of Si B NT. The isodensity surface of the HOMOs of Si N NT is different from that of Si B model so that it is localized on B–B, two Si–B bonds of the SW defect region.

Figure 3
figure 3

Isodensity surfaces of HOMO (left side) and LUMO (right side) for a SW-BNNT, b Si B , c Si N and d Si N.B at the isovalue of 0.02 au. The N, B and Si atoms are represented by blue, pink and gray spheres

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The values of global indices and electric dipole moment for SW defective BNNT and Si-doped SW defective BNNTs before are tabulated in Table 3. One can see that the value of chemical potential, µ, in SiB,N model (−4.88 eV) is smaller than that of SW defective BNNT (−4.01 eV), Si B (−4.00 eV) and Si N (−4.53 eV) models. Decrease in µ upon Si → N substitution in Si N is greater than Si → B substitution in Si B . In Table 3, the Si N,B model has the minimum value of chemical hardness, η, and maximum values of chemical softness, S, as well as electrophilicity index, ω, compared with the other Si-doped SW defective BNNTs. The dipole moment of Si N,B model is smaller than Si B and Si N NTs. There is a correlation between η and HOMO–LUMO gap; its value decreases as the HOMO–LUMO gap decreases. Therefore, reactivity of Si N,B is greater than others.

Table 3 Values of L–H gap, electronic chemical potential (µ), hardness (η), softness (S), electrophilicity index (ω), and dipole moment (Q) for D–N models
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Densities of states (DOSs) can be used as a valuable tool for analyzing the nature of electronic structure. Si atom has the different number of valence electrons than the atoms of the B and N. Accordingly, when a B (or N) atom is substituted by a Si atom, the extra valence electron (or hole) makes available defect levels within the HOMO–LUMO gap of SW-BNNT. Total and partial DOSs of Si B , Si N and Si N,B NTs are shown in Fig. 4. As can be seen, the band gaps of Si-doped SW-BNNTs depend on the positions where the Si atom is substituted. As we know, all electrons in a hexagonal structure of SW-BNNT Si N,B NT are paired and the structure should not present spin polarization while the presence of a Si atom in Si B and Si N NTs causes a net spin polarization. In fact, the unpaired electron in the Si B and Si N NTs induces a magnetic moment.

Figure 4
figure 4

Density of states (DOS) of SW-BNNT, Si N , Si B and Si N,B models. Black curves indicate the total DOS, whereas red, blue, and yellow curves indicate N, B and Si partial DOSs, respectively. Black vertical dashed lines mark the Fermi levels

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Because of the presence of Si atom, DOS curves of Si B , Si N and Si N,B NTs show impurity states in HOMO–LUMO gap. It is clear that the Si impurity has a significant contribution to the DOS appeared in the band gap, so Si doping of BNNT improves the electronic transport property of the SW-BNNT. Compared with SW-BNNT, Fermi level in Si B , Si N and Si N,B NTs moves toward greater energy. As can be seen, Si doping in SiB causes that the new donor-like impurity state to be appear near the conduction band edge and at the same time the new peak appears on the top of the valence band compared with the SW-BNNT, indicating that the SiB NT is an n-type semiconductor with donor impurity states, in good agreement with results found for SiB-BNNT system [27, 67]. While for the electron poor Si N NT, only a new local energy level appears on the top of the valence band, which indicates that Si doping in Si N leads to creating an electronic hole and, in turn, increase in the conductivity of the BNNT. This fact implies that the Si N NT is a typical p-type semiconductor. In Si N,B , two new states are separated by about 4.85 eV.

Effect of axial strain on the geometric and energetic properties of Si-doped (6,0) SW defective BNNTs

The geometric and energetic properties of Si-doped SW defective BNNTs can be affected by axial strain. The change in structural parameters of mentioned NTs upon axial strain is given in Fig. 5. In addition, schematic representation of percentage change in bond lengths upon axial strain is given in Fig. 6. During the axial strain process in Si B model, L 8–13 bonds are stretched up to 5%, L 1, L 3 and L 5 as well as L 15–L 18 bonds are elongated up to 10% and L 2, L 4 and L 6 bonds are lengthened up to 25%. The L 7, L 14 and L 19 bonds decrease under axial strain. The elongation of L 6 bond is greatest, and L 10 bond is smallest in Si B .

Figure 5
figure 5

Change in L 1–L 19 bond lengths upon axial strain in Si N , Si B and Si N,B NTs

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Figure 6
figure 6

Percent of change in L 1–L 19 bind lengths in Si N , Si B and Si N,B NTs upon axial strain

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Under the axial strain in Si N NT, L 7–10 and L 12–14 as well as L 17 bonds are lengthened up to 5%, L 1–L6 and L 15– L 16 as well as L 18 bonds are stretched up to 15% and L 11 and L 19 bonds is shortened; so that the greatest and smallest stretching is related to L 5 and L 10, respectively. The vertical L 19 bond at the 7–7 ring fusion is decreased under axial strain.

In Si N and Si B NTs, the rapid change in bond lengths was not observed, but the changes in L 18 bond length in Si N,B are suddenly so that percent of the changes observed for this bond is 56% under strain employed. The L 10–L 15 are lengthened up to 5%, L 2, L 3, L 5, L 6, L 8 and L 19 bonds are stretched up to 10%, L 1 and L 4 as well as L 16 bonds are elongated up to 25%, and L 7, L 9 and L 17 bonds is shortened upon the vertical strain. In contrast to that seen for vertical L 19 bond localized between two 7–7 rings in Si N and Si B , this bond in Si N,B is lengthened upon the strain. In addition, the most sensitive NT against axial strain is Si N,B model. The predicted fracture progression in Si N , Si B and Si N,B NTs versus axial strain is shown in Fig. 7.

Figure 7
figure 7

Predicted fracture progression in Si N , Si B and Si N,B NTs versus axial strain. The B, N and Si atoms are represented in pink, blue and silver, respectively

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The change in the sum of three bond angles around the N (A, B and C) and B (D, E and F) atoms of vertical L 19 bond in un-doped SW and corresponding atoms in Si-doped NTs Si N , Si B and Si N,B is given in Fig. 8. As can be seen, A and F angles during the axial strain increase and B, C, D and E angles decrease in both Si N and Si B NTs. The average value of the A, B and C decreases and that of D, E and F increases upon axial strain in both NTs. The situation in Si N,B is different. In this NT, C and A angles increase, D and B ones decrease and E as well as F angles do not change nearly upon employing strain. The average value of the A, B and C angles at first decreases and then increases in Si N,B .

Figure 8
figure 8

Changes in A–F angles upon axial strain in Si N , Si B and Si NB NTs (left). Variation of the average value of angles versus axial strain (right)

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The change of radial buckling (β) upon axial strain is given in Fig. 9. This figure shows that the mean values of β for all three doped NTs are positive during the applied axial strain, indicating that the N atoms sit on the outer surface. However, the difference in the radii of the B and N cylinders (β value) of the Si B model first decreases and then increases upon elongation of the NT.

Figure 9
figure 9

Change in average value of radial buckling (β) under axial strain

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The change in the BE and formation energies of Si-doped Si N , Si B and Si N,B NTs are given in Fig. 10. As can be seen, the BE value slightly decreases and formation energy increases when the NTs are stretched. The exceptional resistance of nanotubes to failure is ultimately due to the inherent strength of the constituent chemical bonds and their nearly perfect organization over the microns of the length of the rolled graphite sheet [33]. The elastic strain engineering is one of the nonchemical tools for tuning the electronic properties of nano materials. As expected, since NTs are stretched during axial strain, it is clear that the BE of NTs decreases. Nevertheless, based on the type of NT, their failure is predicted to occur at significantly larger mechanical load. Our results (Fig. 10) show that after 15% tensile strain, BE decreases only ~4.5%. Therefore, the stability of NTs slightly decreases upon axial strain applied and this does not mean that the NT is energetically unstable for further applications. We have checked elasticity of NTs by removing limitations employed during axial strain. After elimination of limitations in NTs, strained NTs are converted to unstrained NT. Although stability of stretched NTs is slightly smaller than unstrained ones, they can retain enough their stability; as the strained BNNTs experimentally have been utilized for different applications such as gas storage [68].

Figure 10
figure 10

Variations of binding energy (left) and defect formation energy (right) against ε

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The change in E HOMO, E LUMO and E L–H as functions of the axial strain in Si-doped SW defective BNNTs is also plotted in Fig. 11. As can be seen, upon axial strain, the E HOMO of Si N and Si B models increases, whereas E HOMO of Si N,B model first increases, then decreases and after that raises again. However, an unlike behavior is observed for E LUMO of models, so that the E LUMO of Si N and Si B decreases and that of Si N,B first rises and then decreases. Figure 11c shows that the energy gap (∆E L–H) of Si N and Si B models decreases continuity and that of Si N,B model first smoothly decreases, then suddenly increases and next reduces again. As can be seen, increase in the energy gap of Si N,B does not reach the initial value of energy gap Si N and Si B models. Thus, it can be predicted that the conductivity of Si N and Si B models increases upon the axial strain and for Si N,B model first increases slowly, then decreases and finally increases.

Figure 11
figure 11

Variations of a HOMO energy (∆E HOMO), b LUMO energy (∆E LUMO) and c HOMO–LUMO gap (∆E L–H) of (6,0) Si-doped SW–BNNTs under axial strain

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Variation of chemical potential (∆µ), chemical hardness (∆η), electrophilicity index (∆ω), chemical softness (∆S) and dipole moment (∆Q) of Si-doped Si N , Si B and Si N,B NTs upon axial strain are given in Fig. 12. As shown, µ value of all NTs first increases during axial strain up to ε ≈ 0.11 and then decreases, in contrast to those of observed for SW defective BNNT [34]. The increase in µ upon axial strain for Si N,B NT is greater than others. The results show that the chemical hardness for Si N and Si B decreases while for Si N,B NT first slowly decreases, then increases and finally decreases, in good agreement with those of observed for electrophilicity index and chemical softness. Therefore, the reactivity behaviors of Si N , Si B upon axial strain are different from that of Si N,B .

Figure 12
figure 12

Variations of a chemical potential (∆µ), b chemical hardness (∆η), c electrophilicity index (∆ω), d chemical softness (∆S) and e dipole moment (∆Q) of (6,0) Si-doped SW defective BNNTs at different strains

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The BNNTs have polar structures, and their dipole moments are affected by defect type. The values of dipole moments of SW defective BNNTs, Si N , Si B and Si N,B NTs models are 5.88, 6.23, 5.58 and 5.56, respectively, so the Si N model has the most polar structures. The axial strain causes which the values of dipole moments change sharply. From Fig. 12e, it is clear that the dipole moment of NTs increases highly upon the axial strain.

NO adsorption on strained and unstrained Si-doped SW-BNNT

Adsorption on the SiB and SiN SW-BNNT

The NO molecule can be adsorbed on strained and unstrained Si1–2-doped SW-BNNTs through either N or O atoms. To explore the stable adsorption configurations of a NO molecule on Si-doped SW-BNNT, the various possible adsorption geometries are considered. Complexes formed from the attachment of NO to unstrained and strained (with 10% strain) Si-doped SW-BNNTs are shown in Fig. 13. The NO molecule has two orientations with respect to the tube axis. The Si Z -NO and Si Z -ON symbols represent that the NO molecule is attached to Si by N and O atoms, respectively. For each of the Si N and Si B NTs interacting with NO gas, complexes of these two types were found on the potential energy surfaces. The electronic adsorption energies (AEs), HOMO, LUMO and band gap energies obtained at M05-2X/6–31+(d) level of theory for Si N -NO, Si N -ON, Si B -NO1, Si B -NO2, Si NB -NO-1, Si NB -ON1-, Si NB -NO2 and Si NB -O-N2 complexes are given in Table 4.

Figure 13
figure 13

Selected structural parameters of unstrained Si-doped SW-BNNT–NO complexes and 10% strained Si-doped SW-BNNT–NO complexes calculated at ONIOM(M05-2X/6–31+G(d):M05-2X/STO-6G) level. Distances are given in Å

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Table 4 Adsorption energies (E ads), HOMO (E HOMO), LUMO (E LUMO) and band gap energies (E g), the charges on the N and O atoms (q N, q O) and the total NBO charge on the NO molecule (CTNBO) of various configurations of Si-doped SW-BNNT–NO complexes
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During the geometry optimization, Si B -ON is converted to another Si B -NO with different configuration Si B -NO2. Therefore, two types of structures Si B -NO1 and Si B -NO2 are predicted for unstrained Si B -NO that direction of NO in the two structures is different. For 1Si-doped SW-BNNTs, AEs calculated for Si N -NO, Si N -ON, Si B -NO1 and Si B -NO2 complexes with unstrained NTs are −26.42, −1.39, −29.92 and −32.69 kcal mol−1. These results first reveal that adsorption of NO gas on the Si B NTs is energetically more favorable than those of Si N NTs. In other words, an energetically favored site for adsorption of NO gas in Si-doped SW-BNNTs is above the Si → B site (the Si atom doping into the B site). Second, adsorption is energetically much less favorable for the Si N -ON than Si N -NO. In contrast to what we see in the Si N-, adsorption of NO on the Si B from its O head is energetically less favorable than that of its N head. It can be concluded that the energetically most favorable site for NO adsorption arises when the NO is attached to Si B from its N head.

Dispersion-corrected AEs at M05-2X-D3/6–31 ++G(d,p)//M05-2X/6–31+G(d) level of theory were also calculated in order to find the effect of dispersion and improved basis set on the interaction energy between NO and Si-doped BNNT. Dispersion-corrected BEs are −27.00, −2.00, −33.43, −33.34 kcal mol−1, respectively. As can be seen, including the dispersion contribution to energies, AEs increase slightly.

AEs obtained for Si-doped–NO complexes are comparable with those reported in the literature. The adsorption energy (∼−16 kcal mol−1) has been reported for the (8,0) SiCNT-NO complex using PBE method [69]. The AEs reported for interaction of NO with C-doped SW-BNNTs at M06-2X/6-31 ++G(d,p) level of theory are −31.29, −9.63, −33.62 and +6.96 kcal mol−1, respectively [24]. As mentioned above, AEs calculated for Si N -NO, Si N -ON, Si B -NO1 and Si B -NO2 complexes are −26.42, −1.39, −29.92 and −32.69 kcal mol−1, respectively. Therefore, SiB-doped SW-BNNT shows an increased ability for adsorption of NO, as compared to the CB-doped SW-BNNT.

The N–O distance in the free NO gas is 1.145 Å and increases to 1.209, 1.267, 1.207 and 1.208 Å in Si N -NO, Si N -ON, Si B -NO1 and Si B -NO2 complexes, respectively, in good agreement with the order of changes in AEs. The elongation of N–O bond can be attributed to the charge transferred from NT to π* orbital of NO molecule. NBO analysis shows that the charge transfer values are 0.463, 0.571, 0.428 and 0.474 au, respectively, especially for Si N -ON complex with the largest charge transfer of 0.571 au, in good agreement with the order of NO bond distances. Besides, Si–N(O) distance is 1.874, 1.777, 1.866 and 1.866 Å in Si N -NO, Si N -ON, Si B -NO1 and Si B -NO2 complexes, respectively. There is not a direct correlation between AEs and Si–N(O) distance.

Band gap for Si N and Si B is 5.43 and 5.50 eV that increases to 5.61, 5.49, 5.81 and 5.96 eV in Si N -NO, Si N -ON, Si B -NO1 and Si B -NO2 complexes, respectively. The percentage change in the band gap energy of Si N -NO, Si N -ON, Si B -NO1 and Si B -NO2 is 3.3, 0.9, 7.0 and 4.3, respectively. On the other hand, the net charge transfer from the electronic-rich Si-doped NT to the NO molecule is 0.428–0.571 au compared to that of 0.22 and 0.15 au reported for Si B -ON and Si N -ON complexes formed from interaction between NO and Si-doped BNNTs [45]. Therefore, it can be predicted that the Si-doped SW-BNNTs are more sensitive than Si-doped BNNTs toward NO gas. The results indicate that the electronic property of the Si-doped BNNTs changes upon adsorption of the NO molecule, so that these changes for Si B -NO1 and Si B -NO2 complexes is greater than those of Si N -NO and Si N -ON ones, in good agreement with results of the band gap energy. Therefore, the sensitivity of Si B -doped NT toward the NO is remarkably greater than that of Si N one. It can be concluded that Si-doped NT not only can adsorb the NO molecule strongly, but also may sense its presence because of the change of the tube band gap.

To gain deeper insights into the electronic structure of the Si-doped SW-BNNTs, we further assessed their density of states (DOS). The presence of a Si atom in Si N and Si B NTs causes the appearance of an unpaired electron and therefore a net spin moment. Therefore, Si doping results in the spin polarization and induces spontaneous magnetization. To explore whether Si doping can induce the spin polarization, the spin DOSs Si N , Si B , Si N -NO, Si B -NO are calculated and shown in Figs. 14 and 15. From this figure, it can be found that new local states are appearing and spin-up and spin-down DOSs are different in the band gap region, indicating that Si doping results in the spin polarization and induces spontaneous magnetization. As can be seen, upon adsorption of NO, Fermi level in these complexes is shifted toward the valence bond and in turn, band gap increases.

Figure 14
figure 14

Density of states (DOS) of Si N and Si N -NO complexes. Black vertical dashed lines mark the Fermi levels

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Figure 15
figure 15

Density of states (DOS) of Si B and Si B -NO complexes. Black vertical dashed lines mark the Fermi levels

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To the best of our knowledge, interaction between strained BNNTs and diatomic gases is not reported and is explored firstly in this work. Before treating AEs for strained systems, it should be noted that in contrast to unstrained NT, Si B -ON is a stable complex and is not converted to Si B -NO. In other words, although unstrained Si B -ON is unstable, but strained Si B -ON(St) complex is energetically stable. AEs calculated for Si N -NO (St), Si N -ON(St), Si B -NO(St) and Si B -ON(St) complexes after 10% tensile strain at M05-2X/6–31+G(d) level of theory are −24.96, 0.83, −31.72 and −8.65 kcal mol−1, respectively, indicating that AE for Si N -NO(St), Si N -ON(St) and Si B -ON(St) decreases after 10% tensile strain and that of Si B -NO(St) increases slightly. Decrease in AE for Si B -ON(St) is greater than others (24 kcal mol−1). Comparison of AEs reveals that changes in the interaction energy after employing 10% strain for Si N -NO(St), Si N -ON(St) and Si B -ON(St) are not significant.

Dispersion-corrected AEs at M05-2X-D3/6–31 ++G(d,p)//M05-2X/6–31+G(d) level of theory for Si N -NO(St), Si N -ON(St), Si B -NO(St) and Si B -ON(St) complexes after 10% tensile strain were also calculated. Dispersion-corrected BEs are −25.50, 0.25, −32.35, −9.29 kcal mol−1, respectively. As can be seen, AEs slightly increases upon including the dispersion contribution. Comparison of unstrained and strained dispersion-corrected AEs reveals that their values slightly decrease upon the axial strain.

The results given in Table 4 show that band gap energies of Si N -NO(St), Si N -ON(St), Si B -NO(St) and Si B -ON(St) complexes change slightly upon 10% axial strain. The results of the NBO analysis given in Table 4 also show that upon adsorption of NO on the strained NT, a net charge transfer of 0.461, 0.572, 0.487 and 0.600 au occurs from the NT to NO molecule.

DOSs of Si N -NO(St), Si N -ON(St), Si B -NO(St) and Si B -ON(St) complexes are given in Fig. 14. As can be observed, upon adsorption of NO, Fermi level in strained complexes is shifted toward the valence bond and, in turn, band gap increases.

Adsorption of NO on the SiN,B SW-BNNTs

Adsorption of NO on Si N,B -doped SW-BNNTs was also investigated. The optimized structures are given in Fig. 13. In Si N,B SW-BNNT, two vertical N and B atoms were substituted by two Si atoms. Four different configurations were found for NO-Si N,B SW-BNNT complexes. NO molecule can be attached to either Si → N (Si substituted for N) or Si → B (Si substituted for B) from N and O heads. AEs calculated for unstrained Si N,B -NO1, Si N,B -ON1, Si N,B -NO2 and Si N,B -ON2 are −18.78, −33.27, −8.75 and 10.32 kcal mol−1, respectively. It should be noted that in Si N,B -ON1, NO molecule interacts with Si atoms via O and N atoms and a closed structure is formed. The AE for this cycloaddition configuration is greater than other ones, indicating that cycloaddition configuration of NO-Si-doped SW-BNNT complex is more stable than other ones. After Si N,B -ON1 complex, Si N,B -NO1 with AE of −18.78 kcal mol−1 is thermodynamically favored. The AE energy of the Si N,B -ON2 complex in which NO interacts from its O head is positive, suggesting that the formation of Si–O bond in this complex is not thermodynamically favored. These results generally propose that Si N,B NT can be practically useful for the abstraction of NO molecules.

In order to estimate the magnitude of the dispersion interaction, single-point calculations were carried out using the M05-2X-D3/6-31 ++G(d,p)//M05-2X/6–31+G(d) level of theory. Dispersion-corrected AEs for Si N,B -NO1, Si N,B -ON1, Si N,B -NO2 and Si N,B -ON2 are −19.29, −33.68, −9.22 and 9.85 kcal mol−1, respectively. As can be observed, the change in AEs after dispersion correction is small.

As NO interacts with Si N,B SW-BNNT, the N–O bond is significantly lengthened. The N–O distance in free NO is 1.145 Å and increases to 1.209, 1,398, 1.214 and 1.310 Å, respectively. As can be seen, a more significant lengthening of NO bond is observed in most stable Si N,B -ON1 complex. The Si–X (X = N, O or NO) distance in unstrained Si NB -NO1, Si N,B -ON1, Si N,B -NO2 and Si N,B -ON2 complexes is 1.875, 1.744 for Si–N (1.853 for Si–O), 1.785 and 1.721 Å, respectively. The Si–Si bond length in Si N,B SW-BNNT is 2.245 Å that increases to 2.311, 2.300 and 2.301 Å in Si N,B -NO1, Si N,B -NO2 and Si NB -ON2 complexes, respectively, with the exception of that of found in most stable Si N,B -ON1 complex that decreases to 2.226 Å.

NBO analysis shows charge is transferred from NT to NO molecule upon complex formation. The net charge transferred is 0.471, 1.035, 0.506 and 0.631 au, respectively, indicating that charge transfer in most stable complex Si N,B -ON1 NT is greater than other complexes.

The band gap in Si N,B is predicted to be 4.85 eV that increases to 5.39, 6.35, 4.96 and 5.36 eV in Si N,B -NO1, Si N,B -ON1, Si N,B -NO2 and Si N,B -ON2 complexes, respectively. The percentage change in the band gap energy is 11.1, 30.9, 2.3 and 10.5, respectively. Therefore, a primary advantage of the change in the band gap of Si N,B NT upon adsorption is the sensitivity of the 2Si-doped SW-BNNTs toward NO gas. The amount of sensitivity depends on position of NO gas above the NT, and its value is greatest when a closed configuration with the structure of Si N,B -ON1 is formed. In other words, results of adsorption of NO onto Si N,B reveal that the Si–Si bond in 2Si-doped SW-BNNTs is highly sensitive to the NO molecule.

To better understand the electronic properties of the complexes, DOSs of unstrained Si N,B -NO1, Si N,B -ON1, Si N,B -NO2 and Si N,B -ON2 complexes were calculated (Fig. 16). As can be seen, Si N,B NT is a nonmagnetic system, whereas complexes formed from the interaction of NO with NT have magnetic property. Change of magnetic properties upon adsorption of NO can be chiefly used for sensing the amount of NO molecules adsorbed on Si N,B NT.

Figure 16
figure 16

Density of states (DOS) of Si N,B and Si N,B -NO complexes. Black vertical dashed lines mark the Fermi levels

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Effects of axial strain on the sensitivity of Si N,B to NO gas are also explored. The most stable structures after adsorption of NO on 10% strained Si N,B are given in Fig. 13. Four complexes named as Si N,B -NO1(St), Si N,B -ON1(St), Si N,B -NO2(St) and Si N,B -ON2(St) complexes were found from interaction between Si N,B and NO gas. The results given in Table 4 show that the NO adsorption properties on the SW-BNNT can be influenced when the B and N atoms of the vertical bond in SW region were substituted with two Si atoms. As can be observed, adsorption energies assessed for Si N,B -NO1(St), Si N,B -ON1(St), Si N,B -NO2(St) and Si N,B -ON2(St) complexes are −15.73, −27.14, −19.37 and 12.52 kcal mol−1, respectively, indicating that, with the exception of formation of Si N,B -ON2(St), adsorption process for other complexes is thermodynamically favored. Comparison of the AEs of strained and unstrained complexes reveals that their values for strained complexes are smaller than unstrained ones, with the exception of Si N,B -NO2 in which AE for it increases from −8.73 to −19.37 kcal mol−1 upon the axial strain.

Dispersion-corrected AEs for Si N,B -NO1(St), Si N,B -ON1(St), Si N,B -NO2(St) and Si N,B -ON2(St) complexes at M05-2X-D3/6-31 ++G(d,p)//M05-2X/6–31+G(d) level of theory are −15.73, −27.14, −19.37 and 12.52 kcal mol−1, respectively, indicating that the adsorption process for formation of strained complexes is thermodynamically favored, with the exception of Si N,B -ON2(St).

The N–O bond is significantly lengthened upon interaction between 10% strained Si NB SW-BNNT and NO gas. The N–O distance in free NO is 1.145 Å and increases to 1.213, 1,398, 1.206 and 1.291 Å in Si N,B -NO1(St), Si N,B -ON1(St), Si N,B -NO2(St) and Si N,B -ON2(St) complexes, respectively. As can be seen, same as to that observed in unstrained Si N,B -ON1 complex, a more significant increase in the NO bond length is observed in most stable Si N,B -ON1(St) complex. The Si–X (X = N, O or NO) distance in strained complexes is 1.845, 1.745 for Si–N (1.856 for Si–O), 1.884 and 1.742 Å, respectively. Comparison of Si–X distances in strained and unstrained complexes shows that the intermolecular distance is lengthened when strained NT interacts with NO gas, with the exception of that observed in Si N,B -NO1(St) that this distance decreases. The Si–Si bond length in Si N,B SW-BNNT is 2.245 Å that increases to 2.286, 2.281 and 2.276 Å in Si N,B -NO1(St), Si N,B -NO2(St) and Si N,B -ON2(St) complexes, with the exception of that of found in the most stable Si N,B -ON1(St) complex that decreases to 2.207 Å.

The results listed in Table 4 demonstrate that band gap energies of strained Si N,B -NO1(St), Si N,B -ON1(St), Si N,B -NO2(St) and Si N,B -ON2(St) are 5.20, 5.86, 5.52 and 5.58 eV, respectively. As can be seen, band gap energies for Si N,B -NO1(St), Si N,B -ON1(St) complexes decrease and those of Si N,B -NO2(St) and Si N,B -ON2(St) ones increase compared with unstrained complexes. The results of the NBO analysis given in Table 4 show that upon adsorption of NO on the strained NT, a net charge of 0.490, 1.044, 0.196 and 0.010 au is transferred from the NT to NO molecule. There is a correlation between band gap energies and charge transfer values so that decrease/increase in band gap energies upon axial strain is accompanied with increase/decrease in charge transfer values.

To gain a better insight about the electronic properties of the complexes, DOSs of unstrained Si N,B -NO1(St), Si N,B -ON1(St), Si N,B -NO2(St) and Si N,B -ON2(St) complexes are calculated and given in Fig. 16. As can be seen, DOS of strained complexes is same as the unstrained complexes. Strained Si N,B NT is a nonmagnetic system and is converted to magnetic systems when interacts with the NO gas. Change of magnetic properties upon the interaction of NO can be principally used for sensing the amount of NO molecules adsorbed on strained Si N,B NT.

Conclusions

The M05-2X density functional theory was employed to investigate the effect of the Si doping defect on the properties of (6,0) SW-BNNT under the axial strain. The calculated binding energy for Si-doped SW defective BNNTs is smaller than SW defective BNNT. In Si-doped SW–BNNT, the defect region serves as a nucleation site for fraction. In general, order of stability of Si-doped SW defective BNNTs is Si N  > Si B  > Si N,B . During axial strain, the formation energy of these NTs increases. The band gap energies, dipole moment, chemical potential, chemical hardness and softness, electrophilicity index, radial buckling and the maximum amount of electronic charge for Si-doped SW–BNNTs were also calculated. Order of energy gap is SW-BNNT > Si B  > Si N  > Si N,B . From a comparison of energy gaps, it can be concluded that the conductivity of Si B,N NT is greater and excitation energy of Si B,N NT is smaller than other two NTs. The band gap decreases upon axial strain for Si B and Si N models, and it almost remains constant in Si N,B model in low tensile strain.

In the second part of this work, sensitivity of strained and unstrained Si-doped SW-BNNTs toward NO gas is evaluated. The results show that the chemical adsorption of NO in many states is thermodynamically favored in both strained and unstrained Si-doped NTs. Thus, it is expected that Si-doped SW-BNNTs could be a favorable gas sensor for sensing the NO molecule.