**2. SWBNNTs under tension, torsion and flattening**

#### **2.1. Simulation procedure**

This section focuses on (*n*,0) SWBNNTs with *n*= 6, 8, 10, where (*n*, *m*) is the chiral index.The (*n*, *m*) tube has a diameter of 3*a n*<sup>2</sup> + *nm* + *n*<sup>2</sup> / *π*, where *a* is the nearest interatomic distance between boron and nitrogen atoms. Figure 1 shows the simulation model of the (8, 0) SWBNNT. The BNNT is located at the center of the unit cell so that the axial direction is parallel to the *z* direction. The cell size is ( 3*an* / *π* + 2*L*V) ×( 3*an* / *π* + 2*L*V) × 3*a* with *a*=0.145 nm and *L*V=0.5 nm, where *L*<sup>V</sup> is the length of vacuum region in the unit cell. Although a three‐dimen‐ sional periodic boundary condition is used, the cell sizes in the *x* and *y* directions are suffi‐ ciently large to avoid any interaction with neighboring image cells. It is confirmed that they have little effect (less than 1%) on the total energy, charge distribution, and energy‐band structure of a deformed BNNT when they are larger than the tube diameter by 1.0 nm.

**Figure 2.** Schematics of tension, torsion, and flattening of SWBNNT.

In tension (Figure 2(a)), the axial strain, *εzz*, is defined as

0 0 *z Z*

(1)

Electronic Structure Calculations for Nano Materials 169

*ϕ<sup>i</sup>* =*θzi* (2)

*Z L L L*

where *Lz0* and *Lz* are the cell sizes in the *z* direction of unstretched and stretched BNNTs,

*zz*

respectively. In this study, *εzz* is in the range 0.00‐0.10 with an increment of 0.02.

In torsion (Figure 2(b)), atom *i* is rotated *ϕ<sup>i</sup>* degrees about the *z*‐axis. *ϕ<sup>i</sup>* is defined as

Atomic positions andthe cell size in the *z*direction are firstrelaxedusing the conjugate gradient method until atomic forces and the stress component, *σzz*, become less than 0.01 eV/Å and 0.01 GPa, respectively. After obtaining the equilibrium structure, tension, torsion, or flattening deformation is applied where atomic configurations are relaxed until their forces become less than 0.01 eV/Å.

**Figure 2.** Schematics of tension, torsion, and flattening of SWBNNT.

stress or by a substrate constraint). Alternatively, BNNTs can be used in nanoelectronic devices by introducing deformation. In any case, our aim is to elucidate the electronic structures of deformed BNNTs. In the following sections, electronic structure calculations for SWBNNTs under tension, torsion, and flattening (Section 2) and for MWBNNTs under flattening (Section

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This section focuses on (*n*,0) SWBNNTs with *n*= 6, 8, 10, where (*n*, *m*) is the chiral index.The (*n*, *m*) tube has a diameter of 3*a n*<sup>2</sup> + *nm* + *n*<sup>2</sup> / *π*, where *a* is the nearest interatomic distance between boron and nitrogen atoms. Figure 1 shows the simulation model of the (8, 0) SWBNNT. The BNNT is located at the center of the unit cell so that the axial direction is parallel to the *z* direction. The cell size is ( 3*an* / *π* + 2*L*V) ×( 3*an* / *π* + 2*L*V) × 3*a* with *a*=0.145 nm and *L*V=0.5 nm, where *L*<sup>V</sup> is the length of vacuum region in the unit cell. Although a three‐dimen‐ sional periodic boundary condition is used, the cell sizes in the *x* and *y* directions are suffi‐ ciently large to avoid any interaction with neighboring image cells. It is confirmed that they have little effect (less than 1%) on the total energy, charge distribution, and energy‐band structure of a deformed BNNT when they are larger than the tube diameter by 1.0 nm.

Atomic positions andthe cell size in the *z*direction are firstrelaxedusing the conjugate gradient method until atomic forces and the stress component, *σzz*, become less than 0.01 eV/Å and 0.01 GPa, respectively. After obtaining the equilibrium structure, tension, torsion, or flattening deformation is applied where atomic configurations are relaxed until their forces become less

**2. SWBNNTs under tension, torsion and flattening**

3) will be discussed.

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**Figure 1.** Simulation model of (8,0) SWBNNT.

**2.1. Simulation procedure**

than 0.01 eV/Å.

In tension (Figure 2(a)), the axial strain, *εzz*, is defined as

$$
\sigma\_{zz} = \frac{L\_z - L\_{Z0}}{L\_{Z0}} \tag{1}
$$

where *Lz0* and *Lz* are the cell sizes in the *z* direction of unstretched and stretched BNNTs, respectively. In this study, *εzz* is in the range 0.00‐0.10 with an increment of 0.02.

In torsion (Figure 2(b)), atom *i* is rotated *ϕ<sup>i</sup>* degrees about the *z*‐axis. *ϕ<sup>i</sup>* is defined as

$$
\varphi\_l = \Theta z\_l \tag{2}
$$

$$\theta = \frac{360N\_{\theta}}{nN\_{\text{z}}L\_{\text{z}0}} \tag{3}$$

deformation modes decrease energy gaps, *Eg = ECBM ‐ EVBM*, of SWBNNTs, but torsion has less

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**Figure 3.** Change in the band structure of an (8,0) SWBNNT. The origin of the energy scale is set at the Fermi level.

of an effect on the energy gap than tension and flattening.

where *θ* is the specific angle of twist, *zi* is the *z*‐coordinate of atom *i*, *N<sup>θ</sup>* is an integer, and *Nz* is the number of primitive unit cells in the *z* direction. The torsion angle must be an integral multiple (*Nθ*) of 360/*n* to fulfill the periodic boundary condition. In this study, the value of *N<sup>θ</sup>* is 1 and that of *Nz* is in the range 3‐5.

In flattening (Figure 2(c)), compression in the *x* direction is applied by reducing the distance between imaginary walls. Once an atom contacts a wall, the atom is allowed to move only on the wall. The flattening ratio, *η*, is defined as

$$
\eta = \frac{D\_0 \cdot D}{D\_0} \tag{4}
$$

where *D0* is the tube diameter at equilibrium and *D* is the distance between the imaginary walls. In this study, *η* is in the range 0.00‐0.50 with an increment of 0.05.

First‐principles DFT calculations are conducted using the Vienna *ab initio* Simulation Package (VASP) [32, 33]. The wave functions are expanded in a plane‐wave basis set with a cut‐off energy of 350 eV. The ultrasoft pseudopotential proposed by Vanderbilt [34] is used and the exchange‐correlation energy is evaluated by the generalized gradient approximation of Perdew and Wang [35]. The Brillouin zone integration is performed by the Monkhorst‐Pack scheme [36] using a 1 × 1 × 4 *k*‐point mesh for atomic and electronic relaxations. After the relaxation, the energy‐band structure is obtained by calculating energy eigenvalues of 30 points on Γ‐Χ line in the Brillouin zone.

#### **2.2. Results and discussion**

#### *2.2.1. Energy‐band structures*

It is well known that the DFT underestimates the energy gap. For a quantitative discussion of the energy gap, a modified theory such as the GW approximation (GWA) is necessary [37‐40]. Nonetheless, previous studies on bulk hexagonal BN and an isolated BN sheet showed that the shape of the energy bands by the DFT is quite similar to that by the GWA except for the magnitude of the energy gap [37, 38]. Thus, the DFT can qualitatively predict energy‐band structures of BNNTs.

Figure 3 shows the change in the energy‐band structures of the (8,0) SWBNNT under tension, torsion, and flattening. The (6,0) and (10,0) show a changing trend similar to the (8,0) band structure. A common feature among the three deformation modes is that both the valence‐ band maximum (VBM) and conduction‐band minimum (CBM) are located at the Γ point (*k*=0) during the deformations. Another common feature is that the change in the energy of the VBM, *EVBM*, is almost zero. Note that while the tension and flattening obviously decrease the energy of the CBM, *ECBM*, the torsion hardly decreases *ECBM*. The results suggest that all the three deformation modes decrease energy gaps, *Eg = ECBM ‐ EVBM*, of SWBNNTs, but torsion has less of an effect on the energy gap than tension and flattening.

0

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

where *θ* is the specific angle of twist, *zi* is the *z*‐coordinate of atom *i*, *N<sup>θ</sup>* is an integer, and *Nz* is the number of primitive unit cells in the *z* direction. The torsion angle must be an integral multiple (*Nθ*) of 360/*n* to fulfill the periodic boundary condition. In this study, the value of *N<sup>θ</sup>*

In flattening (Figure 2(c)), compression in the *x* direction is applied by reducing the distance between imaginary walls. Once an atom contacts a wall, the atom is allowed to move only on

where *D0* is the tube diameter at equilibrium and *D* is the distance between the imaginary

First‐principles DFT calculations are conducted using the Vienna *ab initio* Simulation Package (VASP) [32, 33]. The wave functions are expanded in a plane‐wave basis set with a cut‐off energy of 350 eV. The ultrasoft pseudopotential proposed by Vanderbilt [34] is used and the exchange‐correlation energy is evaluated by the generalized gradient approximation of Perdew and Wang [35]. The Brillouin zone integration is performed by the Monkhorst‐Pack scheme [36] using a 1 × 1 × 4 *k*‐point mesh for atomic and electronic relaxations. After the relaxation, the energy‐band structure is obtained by calculating energy eigenvalues of 30

It is well known that the DFT underestimates the energy gap. For a quantitative discussion of the energy gap, a modified theory such as the GW approximation (GWA) is necessary [37‐40]. Nonetheless, previous studies on bulk hexagonal BN and an isolated BN sheet showed that the shape of the energy bands by the DFT is quite similar to that by the GWA except for the magnitude of the energy gap [37, 38]. Thus, the DFT can qualitatively predict energy‐band

Figure 3 shows the change in the energy‐band structures of the (8,0) SWBNNT under tension, torsion, and flattening. The (6,0) and (10,0) show a changing trend similar to the (8,0) band structure. A common feature among the three deformation modes is that both the valence‐ band maximum (VBM) and conduction‐band minimum (CBM) are located at the Γ point (*k*=0) during the deformations. Another common feature is that the change in the energy of the VBM, *EVBM*, is almost zero. Note that while the tension and flattening obviously decrease the energy of the CBM, *ECBM*, the torsion hardly decreases *ECBM*. The results suggest that all the three

*<sup>η</sup>* <sup>=</sup> *<sup>D</sup>*<sup>0</sup> ‐ *<sup>D</sup> D*0

walls. In this study, *η* is in the range 0.00‐0.50 with an increment of 0.05.

(3)

(4)

360 *z z N nN L* 

is 1 and that of *Nz* is in the range 3‐5.

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the wall. The flattening ratio, *η*, is defined as

points on Γ‐Χ line in the Brillouin zone.

**2.2. Results and discussion**

*2.2.1. Energy‐band structures*

structures of BNNTs.

**Figure 3.** Change in the band structure of an (8,0) SWBNNT. The origin of the energy scale is set at the Fermi level.

**Figure 4.** Energy gaps of the SWBNNTs as a function of (a) axial strain, (b) specific angle of twist, and (c) flattening ratio.

Figure 4 shows the energy gaps of the (6,0), (8,0), and (10,0) SWBNNTs as a function of the axial strain, specific angle of twist, and flattening ratio. The energy gap of the (10,0) with *θ*=41.7 deg/nm (*Nz*=5)is not shown in the figure because it collapsed. Undertension and torsion except for *θ*=20.8‐27.8 deg/nm in the (10,0), the energy gap decreases almost linearly and the rate of decrease hardly depends on the diameter. In contrast, under flattening, the energy gap decreases quadratically or exponentially and the amount of decrease significantly depends on the diameter; a SWBNNT with the smaller diameter shows a larger decrease in the energy gap. It is also shown that flattening results in a few times larger decrease in the energy gap than tension and torsion.

state along a circumference passing through boron atoms under no deformation (Figure 6(a), *εzz*=0.00), and that the tension, torsion, and flattening induce the change in the CBM state.

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**Figure 5.** Change in the energy gap of (8,0) SWBNNT under flattening following axial tension.}

With increasing flattening deformation (Figure 6(d)), charges are transferred from the flattened to the curved regions, resulting in an overlap of the charge densities and formations of electronic bonds between neighboring boron atoms in the curved regions. It is this mechanism thatresults in the decrease in *ECBM* in the flattened SWBNNTs. Comparing the three SWBNNTs with *η*=0.45 (Figures 6(d)‐(f)), the electronic bonds become stronger as the diameter becomes smaller. Therefore, a flattened SWBNNT with a smaller diameter shows a larger decrease in

Under tension (Figure 6(a)), the tube curvature increases because of Poisson contraction, leading to the enhancement of *π\*‐σ\** hybridizations and the decrease in *ECBM*. Figure 6(a) shows the narrowing white center area of zero‐charge densities and the spreading gray area of *π\*‐ σ\** hybridizations. The same is true for the torsion (Figure 6(b)), but it induces less change in charge densities than tension (the size of the white center area changes little in Figure 6(b)), resulting in a smaller decrease in the energy gap under torsion than under tension (Figures 4(a),(b)). It should be noted that elastic buckling occurred at a *θ* between 20.8 and 27.8 deg/nm in the (10,0), leading to local flattening (Figure 6(c)). Therefore, the relation of *Eg* versus *θ* deviates from the linear decrease at *θ* of 20.8‐27.8 deg/nm in the (10,0)(Figure 4(b)).Itis obvious that the overlap of charge densities is much stronger under flattening than under tension or torsion. Therefore, the decrease in the energy gap in the formeris much largerthan in the latter.

the energy gap.

Although the discussion so far in this section has dealt with the SWBNNTs under the three simple deformation modes, BNNTs would be subjected to combined deformation in their practical use. Therefore, the energy gap of the SWBNNTs subjected to flattening following axial tension is further analyzed (Figure 5). It is found that preceding tension shifts an *Eg‐η* curve downward without dramatic changes in its shape, and that the extent of the shift almost corresponds to the energy gap decrease induced by simple tension (Figure 4(a)). This result suggests that the energy gap of the SWBNNTs under a combination of the three deformation modes can be deduced from Figure 4. In the rest of this section, therefore, only the simple deformation modes will be discussed.

#### *2.2.2. Charge densities at the CBM*

Here the mechanism of deformation‐induced electronic changes in the SWBNNTs is discussed in terms of charge densities at the CBM (Figure 6). The CBM is composed of boron‐derived states. In fact, CBM charge densities are high around boron atoms, while they are low around nitrogen atoms. It is found that the *π\** state (*pz* orbitals of boron atoms) hybridizes with the *σ\**

**Figure 5.** Change in the energy gap of (8,0) SWBNNT under flattening following axial tension.}

**Figure 4.** Energy gaps of the SWBNNTs as a function of (a) axial strain, (b) specific angle of twist, and (c) flattening

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

Figure 4 shows the energy gaps of the (6,0), (8,0), and (10,0) SWBNNTs as a function of the axial strain, specific angle of twist, and flattening ratio. The energy gap of the (10,0) with *θ*=41.7 deg/nm (*Nz*=5)is not shown in the figure because it collapsed. Undertension and torsion except for *θ*=20.8‐27.8 deg/nm in the (10,0), the energy gap decreases almost linearly and the rate of decrease hardly depends on the diameter. In contrast, under flattening, the energy gap decreases quadratically or exponentially and the amount of decrease significantly depends on the diameter; a SWBNNT with the smaller diameter shows a larger decrease in the energy gap. It is also shown that flattening results in a few times larger decrease in the energy gap than

Although the discussion so far in this section has dealt with the SWBNNTs under the three simple deformation modes, BNNTs would be subjected to combined deformation in their practical use. Therefore, the energy gap of the SWBNNTs subjected to flattening following axial tension is further analyzed (Figure 5). It is found that preceding tension shifts an *Eg‐η* curve downward without dramatic changes in its shape, and that the extent of the shift almost corresponds to the energy gap decrease induced by simple tension (Figure 4(a)). This result suggests that the energy gap of the SWBNNTs under a combination of the three deformation modes can be deduced from Figure 4. In the rest of this section, therefore, only the simple

Here the mechanism of deformation‐induced electronic changes in the SWBNNTs is discussed in terms of charge densities at the CBM (Figure 6). The CBM is composed of boron‐derived states. In fact, CBM charge densities are high around boron atoms, while they are low around nitrogen atoms. It is found that the *π\** state (*pz* orbitals of boron atoms) hybridizes with the *σ\**

ratio.

tension and torsion.

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deformation modes will be discussed.

*2.2.2. Charge densities at the CBM*

state along a circumference passing through boron atoms under no deformation (Figure 6(a), *εzz*=0.00), and that the tension, torsion, and flattening induce the change in the CBM state.

With increasing flattening deformation (Figure 6(d)), charges are transferred from the flattened to the curved regions, resulting in an overlap of the charge densities and formations of electronic bonds between neighboring boron atoms in the curved regions. It is this mechanism thatresults in the decrease in *ECBM* in the flattened SWBNNTs. Comparing the three SWBNNTs with *η*=0.45 (Figures 6(d)‐(f)), the electronic bonds become stronger as the diameter becomes smaller. Therefore, a flattened SWBNNT with a smaller diameter shows a larger decrease in the energy gap.

Under tension (Figure 6(a)), the tube curvature increases because of Poisson contraction, leading to the enhancement of *π\*‐σ\** hybridizations and the decrease in *ECBM*. Figure 6(a) shows the narrowing white center area of zero‐charge densities and the spreading gray area of *π\*‐ σ\** hybridizations. The same is true for the torsion (Figure 6(b)), but it induces less change in charge densities than tension (the size of the white center area changes little in Figure 6(b)), resulting in a smaller decrease in the energy gap under torsion than under tension (Figures 4(a),(b)). It should be noted that elastic buckling occurred at a *θ* between 20.8 and 27.8 deg/nm in the (10,0), leading to local flattening (Figure 6(c)). Therefore, the relation of *Eg* versus *θ* deviates from the linear decrease at *θ* of 20.8‐27.8 deg/nm in the (10,0)(Figure 4(b)).Itis obvious that the overlap of charge densities is much stronger under flattening than under tension or torsion. Therefore, the decrease in the energy gap in the formeris much largerthan in the latter.

**Figure 7.** Deformation energy of the SWBNNTs as a function of (a) axial strain, (b) specific angle of twist, (c) flattening

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Figure 9 shows the relationship between energy gap and deformation force. The three bands to the right illustrate the obtainable range of the energy gap by introducing tension, torsion, and flattening. In tension and torsion, a larger force is required for a larger tube to induce the same amount of energy gap decrease. The opposite is true in flattening, i.e., a larger force for a smaller tube. The key findings from Figure 9 are that (i) the flattening with a force smaller than that applied for tension or torsion leads to the larger decrease in the energy gap, and (ii) flattening offers a larger obtainable range of the energy gap than tension and torsion: 1.4‐4.0 eV under flattening, 2.5‐2.8 eV and 3.1‐4.0 eV under tension, 2.7‐2.8 eV, 3.4‐3.6 eV, and 3.8‐4.0 eV under torsion. These findings indicate that flattening has the potential to enable BNNTs to be used as nanoelectronic devices. However, a valid question is whether flattening BNNTs is

**Figure 8.** Forces required to deform (6,0), (8,0) and (10,0) SWBNNTs.

experimentally feasible.

ratio.

**Figure 6.** Change in the CBM charge density. Cross sections passing through boron atoms are shown.

#### *2.2.3. Deformation forces*

Figure 7 shows the deformation energy as a function of axial strain, specific angle of twist, and flattening ratio. The curves in tension, torsion, and flattening are fitted by cubic, quadratic, and quartic polynomials, respectively. The first and second derivatives of each curve provide the deformation force (Figure 8) and the elastic modulus, respectively. Youngʹs moduli of the (6,0), (8,0), and (10,0) are thus calculated to be 0.759, 0.794, and 0.811 TPa, respectively. They are in good agreement with those measured in experiments (1.22±0.24 TPa [10] and 0.722 TPa [16]) and otherfirst‐principles calculations (0.762, 0.785, and 0.803 TPa for(6,0),(8,0), and (10,0), respectively [18]). It is found in Figure 8 that forces under flattening are smaller than under tension and torsion, because strong in‐plane B‐N covalent bonds prevent in‐plane tension and torsion. It is also found that forces rapidly increase later under flattening. The rapid increase starts from around *η*=0.3 and 0.4 in the (6,0) and (8,0), respectively, where the imaginary wall distances are 0.35 and 0.38 nm, respectively. Because the interlayer distance of hexagonal BNs and MWBNNTs is around 0.34 nm, the rapid increase would be attributed to the repulsive force between the two flattened regions.

**Figure 7.** Deformation energy of the SWBNNTs as a function of (a) axial strain, (b) specific angle of twist, (c) flattening ratio.

**Figure 8.** Forces required to deform (6,0), (8,0) and (10,0) SWBNNTs.

*2.2.3. Deformation forces*

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force between the two flattened regions.

Figure 7 shows the deformation energy as a function of axial strain, specific angle of twist, and flattening ratio. The curves in tension, torsion, and flattening are fitted by cubic, quadratic, and quartic polynomials, respectively. The first and second derivatives of each curve provide the deformation force (Figure 8) and the elastic modulus, respectively. Youngʹs moduli of the (6,0), (8,0), and (10,0) are thus calculated to be 0.759, 0.794, and 0.811 TPa, respectively. They are in good agreement with those measured in experiments (1.22±0.24 TPa [10] and 0.722 TPa [16]) and otherfirst‐principles calculations (0.762, 0.785, and 0.803 TPa for(6,0),(8,0), and (10,0), respectively [18]). It is found in Figure 8 that forces under flattening are smaller than under tension and torsion, because strong in‐plane B‐N covalent bonds prevent in‐plane tension and torsion. It is also found that forces rapidly increase later under flattening. The rapid increase starts from around *η*=0.3 and 0.4 in the (6,0) and (8,0), respectively, where the imaginary wall distances are 0.35 and 0.38 nm, respectively. Because the interlayer distance of hexagonal BNs and MWBNNTs is around 0.34 nm, the rapid increase would be attributed to the repulsive

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

**Figure 6.** Change in the CBM charge density. Cross sections passing through boron atoms are shown.

Figure 9 shows the relationship between energy gap and deformation force. The three bands to the right illustrate the obtainable range of the energy gap by introducing tension, torsion, and flattening. In tension and torsion, a larger force is required for a larger tube to induce the same amount of energy gap decrease. The opposite is true in flattening, i.e., a larger force for a smaller tube. The key findings from Figure 9 are that (i) the flattening with a force smaller than that applied for tension or torsion leads to the larger decrease in the energy gap, and (ii) flattening offers a larger obtainable range of the energy gap than tension and torsion: 1.4‐4.0 eV under flattening, 2.5‐2.8 eV and 3.1‐4.0 eV under tension, 2.7‐2.8 eV, 3.4‐3.6 eV, and 3.8‐4.0 eV under torsion. These findings indicate that flattening has the potential to enable BNNTs to be used as nanoelectronic devices. However, a valid question is whether flattening BNNTs is experimentally feasible.

**3. MWBNNTs under flattening**

**Figure 10.** Simulation model of (13,0)@(21,0) DWBNNT.

This section focuses on (5,0), (13,0), and (21,0) SW, (5,0)@(13,0) and (13,0)@(21,0) DW, and (5,0)@(13,0)@(21,0) TWBNNTs. Figure 10 shows the simulation model of the (13,0)@(21,0) DWBNNT. The initial nearest interatomic distance between boron and nitrogen atoms is set as 0.145 nm. Boron (nitrogen) atoms in the outer tube are stacked above nitrogen (boron) atoms in the inner tube [42]. The axial direction of the BNNT is parallel to the *z*‐direction. The BNNT is located at the center of the unit cell with a size of 3.637 nm × 3.637 nm × 0.435 nm. Even though a three‐dimensional periodic boundary condition is employed, the cell sizes in the *x*‐ and *y*‐directions are large enough to avoid interaction with neighboring image cells, because they have little effect (less than 1%) on the energy, charge distribution, and energy band structure of a flattened BNNT, when they are greater than the diameter of the BNNT

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**3.1. Simulation procedure**

plus 1.0 nm.

**Figure 9.** Relationship between energy gap and force of (6,0), (8,0) and (10,0) SWBNNTs.

In order to answer this question, the estimated flattening forces are compared with those of SWCNTs that Barboza et al. have already experimentally succeeded in flattening by means of an atomic force microscopy (AFM) tip [41]. Although they did not actually measure flattening forces of (*n*,0) SWCNTs with *n* ≤ 10, they proposed and validated a universal relationship among the applied force, SWCNT diameter, AFM tip radius, and flattening ratio:

$$\frac{{}^{F}D\_{0}^{\otimes 2}}{(2R)^{3/2}} = \frac{\alpha}{(1-\eta)^{3/2}} \left[ \sqrt{2\eta + \eta^{2}} + t\_{\mathcal{S}} \cdot {}^{1} \left( \sqrt{\frac{\eta}{1-\eta}} \right) \right] \tag{5}$$

where *R* is the AFM tip radius and *α* is a constant (=1.2×10‐<sup>18</sup> J). Equation (5) indicates that the quantity FD0 3/2(2R)‐1/2 should be universal to any SWCNT. They showed that all experimental data fall on a single curve obtained by Equation (5) up to *η* ≅ 0.4. From Equation (5) and the geometric contact conditions between a tube and an AFM tip, the flattening force per unit length of a (6,0) SWCNT (*D0*=0.470 nm) is calculated to be 15.4 N/m when *η*=0.4 and *R*=30 nm. In contrast, from Figure 8, that of the (6,0) SWBNNT (=*F*/*Lz0*) is estimated to be 16.8 N/m at *η*=0.4. The results demonstrate that the flattening force is almost equal in SWCNTs and SWBNNTs, indicating that the same experiments as Barboza et al. would be feasible for SWBNNTs. The fact that CNTs and BNNTs almost have the same tube shape and size when their chiral indexes are the same (*a* ≅ 0.142 nm in CNTs and *a* ≅ 0.145 nm in BNNTs) also encourages the feasibility of flattening BNNTs. It is therefore concluded that the flattening forces estimated are not unrealistic and strongly expected thatthe same or similar experimental technique also applies to BNNTs.
