**3. MWBNNTs under flattening**

#### **3.1. Simulation procedure**

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

*F D*<sup>0</sup> 3/2 (2*R*)1/2 <sup>=</sup> *<sup>α</sup>*

quantity FD0

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technique also applies to BNNTs.

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

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(1 ‐ *<sup>η</sup>*)3/2 <sup>2</sup>*<sup>η</sup>* <sup>+</sup> *<sup>η</sup>* <sup>2</sup> <sup>+</sup> *t g* ‐1( *<sup>η</sup>*

where *R* is the AFM tip radius and *α* is a constant (=1.2×10‐<sup>18</sup> J). Equation (5) indicates that the

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

3/2(2R)‐1/2 should be universal to any SWCNT. They showed that all experimental

<sup>1</sup> ‐ *<sup>η</sup>* ) (5)

among the applied force, SWCNT diameter, AFM tip radius, and flattening ratio:

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 plus 1.0 nm.

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

**Figure 11.** Schematic illustration explaining flattening compression of BNNTs.

Atomic positions and the cell size in the *z*‐direction are first relaxed using 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, a flattening compression in the *x*‐direction is applied by reducing the distance between imaginary walls until the BNNT collapses (Figure 11). Once an atom contacts a wall, the atom is allowed to move only on the wall. During compression, the cell sizes are fixed and atomic configurations are relaxed until their forces become less than 0.01 eV/Å. To investigate the degree of defor‐ mation, the flattening ratio, *η*, is defined as

$$
\eta = \frac{D\_0 \cdot D}{D\_0} \tag{6}
$$

(13,0)@(21,0) DWBNNT are located at the Γ point (*k*=0) during the deformation, but those of the (5,0)@(13,0) DWBNNT move to *k*≠0 midway during the deformation and then return to the Γ point. In each BNNT, the energy of the VBM, *E*VBM, hardly changes, while that of the CBM, *E*CBM, changes, indicating that the change in the energy gap, *Eg*, is mainly caused by a change in *E*CBM. In the (13,0) SW and (13,0)@(21,0) DWBNNTs (Figure 12(a), (b)), *E*CBM decreases monotonically. In contrast, in the (5,0)@(13,0) DWBNNT (Figure 12(c)), *E*CBM first increases and

Electronic Structure Calculations for Nano Materials 179

**Figure 12.** Change in the band structure of (13,0) SWBNNT and (13,0)@(21,0) and (5,0)@(13,0) DWBNNTs in flatten‐

ing deformation. The origin of the energy scale is set at the Fermi level.

then decreases.

where *D0* is the diameter of the outermost tube at equilibrium, and *D* is the distance between the imaginary walls.

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.

#### **3.2. Results and discussion**

#### *3.2.1. Energy‐band structures*

Figure 12 shows the change in the energy band structures of the (13,0) SWBNNT and (13,0)@(21,0) and (5,0)@(13,0) DWBNNTs during flattening deformation. The other SWBNNTs and the (5,0)@(13,0)@(21,0) TWBNNT show a similar changing trend of the band structure to the (13,0) SWBNNT and the (5,0)@(13,0) DWBNNT, respectively. Both the valence band maximum (VBM) and the conduction band minimum (CBM) of the (13,0) SWBNNT and (13,0)@(21,0) DWBNNT are located at the Γ point (*k*=0) during the deformation, but those of the (5,0)@(13,0) DWBNNT move to *k*≠0 midway during the deformation and then return to the Γ point. In each BNNT, the energy of the VBM, *E*VBM, hardly changes, while that of the CBM, *E*CBM, changes, indicating that the change in the energy gap, *Eg*, is mainly caused by a change in *E*CBM. In the (13,0) SW and (13,0)@(21,0) DWBNNTs (Figure 12(a), (b)), *E*CBM decreases monotonically. In contrast, in the (5,0)@(13,0) DWBNNT (Figure 12(c)), *E*CBM first increases and then decreases.

**Figure 11.** Schematic illustration explaining flattening compression of BNNTs.

mation, the flattening ratio, *η*, is defined as

scheme [36] using a 1 × 1× 4 *k*‐point mesh.

**3.2. Results and discussion**

*3.2.1. Energy‐band structures*

the imaginary walls.

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Atomic positions and the cell size in the *z*‐direction are first relaxed using 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, a flattening compression in the *x*‐direction is applied by reducing the distance between imaginary walls until the BNNT collapses (Figure 11). Once an atom contacts a wall, the atom is allowed to move only on the wall. During compression, the cell sizes are fixed and atomic configurations are relaxed until their forces become less than 0.01 eV/Å. To investigate the degree of defor‐

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

where *D0* is the diameter of the outermost tube at equilibrium, and *D* is the distance between

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

Figure 12 shows the change in the energy band structures of the (13,0) SWBNNT and (13,0)@(21,0) and (5,0)@(13,0) DWBNNTs during flattening deformation. The other SWBNNTs and the (5,0)@(13,0)@(21,0) TWBNNT show a similar changing trend of the band structure to the (13,0) SWBNNT and the (5,0)@(13,0) DWBNNT, respectively. Both the valence band maximum (VBM) and the conduction band minimum (CBM) of the (13,0) SWBNNT and

(6)

**Figure 12.** Change in the band structure of (13,0) SWBNNT and (13,0)@(21,0) and (5,0)@(13,0) DWBNNTs in flatten‐ ing deformation. The origin of the energy scale is set at the Fermi level.

> (21,0) SWBNNTs changes from a NFE‐like state to a *π\*‐σ\** hybridized state is that the tube curvature increases locally as the flattening deformation increases (in (*n*,0) SWBNNTs under no deformation, the CBM is a NFE‐like state when *n ≥* 13 and a *π\*‐σ\** hybridized state when *n* < 13, and the hybridization becomes stronger with increasing tube curvature) [8]. The energy gap of the (5,0) SWBNNT is much smallerthan those of the (13,0) and (21,0) SWBNNTs because of its strong *π\*‐σ\** hybridization. With increasing flattening deformation, charge is transferred from flattened regions to curved ones, leading to an overlap of the charge densities. The *E*CBM of the SWBNNTs decreases under flattening because of the formation of electronic bonds between neighboring boron atoms in the curved regions. The charge density distribution in curved regions of the (13,0) SWBNNT at *η*=0.21 is similar to that of the (21,0) SWBNNT at *η*=0.52, which results in them having almost the same energy gap of 4.2 eV. This is because they have almost the same value of *D*, namely the same curvature of the curved region. Figure 15 shows the relationship between the energy gap and imaginary wall distance of the (13,0) and (21,0) SWBNNTs. Their energy gaps are almost equal under a same wall distance.

> > -

-

-

-

-

-

-

**Figure 14.** Change in the CBM charge density of (5,0), (13,0), and (21,0) SWBNNTs in flattening deformation.


-

-

 

-

 

-

Electronic Structure Calculations for Nano Materials 181

-

-

**Figure 13.** Relationship between energy gap, Eg, and flattening ratio, η, of (5,0), (13,0), and (21,0) SWBNNTs, (5,0)@(13,0) and (13,0)@(21,0) DWBNNTs, and (5,0)@(13,0)@(21,0) TWBNNT.

Figure 13 shows the energy gaps of the (5,0), (13,0), and (21,0) SWBNNTs, (5,0)@(13,0) and (13,0)@(21,0) DWBNNTs, and (5,0)@(13,0)@(21,0) TWBNNT as a function ofthe flattening ratio. The energy gap of the three SWBNNTs decreases almost monotonically, and the amount of decrease becomes smaller with increasing tube diameter. The energy gap of the (13,0)@(21,0) DWBNNT also decreases monotonically, but it exhibits a more rapid decrease than the (13,0) and (21,0) SWBNNTs. It should be noted that the energy gaps of the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT increase during the early stage and then decrease. This shift occurs earlier in the latter (*η*=0.34) than in the former (*η*=0.48). The fact that the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT show different changing trends of *Eg* from the SWBNNTs proves that interwall interactions significantly affect the electronic structures of the flattened (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT.

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

Figure 14 shows charge densities at the CBM of the flattened (5,0), (13,0), and (21,0) SWBNNTs at a cross section passing through boron atoms. The characteristics of the nearly free electron (NFE) state are observed in the BNNTs with a small curvature (Figure 14(b): *η*=0.00, (c): *η*=0.00, 0.20), while *π\*‐σ\** hybridizations appear in the others. The reason the CBM of the (13,0) and (21,0) SWBNNTs changes from a NFE‐like state to a *π\*‐σ\** hybridized state is that the tube curvature increases locally as the flattening deformation increases (in (*n*,0) SWBNNTs under no deformation, the CBM is a NFE‐like state when *n ≥* 13 and a *π\*‐σ\** hybridized state when *n* < 13, and the hybridization becomes stronger with increasing tube curvature) [8]. The energy gap of the (5,0) SWBNNT is much smallerthan those of the (13,0) and (21,0) SWBNNTs because of its strong *π\*‐σ\** hybridization. With increasing flattening deformation, charge is transferred from flattened regions to curved ones, leading to an overlap of the charge densities. The *E*CBM of the SWBNNTs decreases under flattening because of the formation of electronic bonds between neighboring boron atoms in the curved regions. The charge density distribution in curved regions of the (13,0) SWBNNT at *η*=0.21 is similar to that of the (21,0) SWBNNT at *η*=0.52, which results in them having almost the same energy gap of 4.2 eV. This is because they have almost the same value of *D*, namely the same curvature of the curved region. Figure 15 shows the relationship between the energy gap and imaginary wall distance of the (13,0) and (21,0) SWBNNTs. Their energy gaps are almost equal under a same wall distance.


*3.2.2. Charge densities at the CBM*



(5,0)@(13,0) and (13,0)@(21,0) DWBNNTs, and (5,0)@(13,0)@(21,0) TWBNNT.

flattened (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT.



 

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**Figure 13.** Relationship between energy gap, Eg, and flattening ratio, η, of (5,0), (13,0), and (21,0) SWBNNTs,

Figure 13 shows the energy gaps of the (5,0), (13,0), and (21,0) SWBNNTs, (5,0)@(13,0) and (13,0)@(21,0) DWBNNTs, and (5,0)@(13,0)@(21,0) TWBNNT as a function ofthe flattening ratio. The energy gap of the three SWBNNTs decreases almost monotonically, and the amount of decrease becomes smaller with increasing tube diameter. The energy gap of the (13,0)@(21,0) DWBNNT also decreases monotonically, but it exhibits a more rapid decrease than the (13,0) and (21,0) SWBNNTs. It should be noted that the energy gaps of the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT increase during the early stage and then decrease. This shift occurs earlier in the latter (*η*=0.34) than in the former (*η*=0.48). The fact that the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT show different changing trends of *Eg* from the SWBNNTs proves that interwall interactions significantly affect the electronic structures of the

Figure 14 shows charge densities at the CBM of the flattened (5,0), (13,0), and (21,0) SWBNNTs at a cross section passing through boron atoms. The characteristics of the nearly free electron (NFE) state are observed in the BNNTs with a small curvature (Figure 14(b): *η*=0.00, (c): *η*=0.00, 0.20), while *π\*‐σ\** hybridizations appear in the others. The reason the CBM of the (13,0) and


 






 

 






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**Figure 14.** Change in the CBM charge density of (5,0), (13,0), and (21,0) SWBNNTs in flattening deformation.

> during deformation, one might expect that the *E*g*‐η* curve of the (13,0)@(21,0) DWBNNT coincides with that of the (13,0) SWBNNT. However, *E*<sup>g</sup> of the former is in fact smaller than that of the latter under the same *η*. As shown in Figure 16, the flattening ratio of the innermost tube, *η*in, must be larger than *η* to maintain the interwall spacing constant. This means that the B‐B bonds in the flattened (13,0)@(21,0) DWBNNT are strongerthan those in the flattened (13,0) SWBNNT under the same *η*, resulting in a larger decrease in *E*CBM in the former than in the

latter.


flattening deformation.

 

 

 

 

 

 

Electronic Structure Calculations for Nano Materials 183

 

!

**Figure 17.** Change in the CBM charge density of (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT during

 

 

 

 " #  

**Figure 15.** Energy gap of (13,0) and (21,0) SWBNNTs as a function of imaginary wall distance.

**Figure 16.** Change in the CBM charge density of (13,0)@(21,0) DWBNNT during flattening deformation.

The CBM charge density distribution of the (13,0)@(21,0) DWBNNT is similar to that of the SWBNNTs (Figure 16). In the inner tube, charge transfer from flattened to curved regions is observed and an overlap of the charge densities is induced in the curved regions. The decrease in *E*CBM of the (13,0)@(21,0) DWBNNT is caused by the same mechanism as in the SWBNNTs mentioned above. Because the charge densities are distributed almost entirely in the innertube during deformation, one might expect that the *E*g*‐η* curve of the (13,0)@(21,0) DWBNNT coincides with that of the (13,0) SWBNNT. However, *E*<sup>g</sup> of the former is in fact smaller than that of the latter under the same *η*. As shown in Figure 16, the flattening ratio of the innermost tube, *η*in, must be larger than *η* to maintain the interwall spacing constant. This means that the B‐B bonds in the flattened (13,0)@(21,0) DWBNNT are strongerthan those in the flattened (13,0) SWBNNT under the same *η*, resulting in a larger decrease in *E*CBM in the former than in the latter.





**Figure 15.** Energy gap of (13,0) and (21,0) SWBNNTs as a function of imaginary wall distance.



**Figure 16.** Change in the CBM charge density of (13,0)@(21,0) DWBNNT during flattening deformation.

The CBM charge density distribution of the (13,0)@(21,0) DWBNNT is similar to that of the SWBNNTs (Figure 16). In the inner tube, charge transfer from flattened to curved regions is observed and an overlap of the charge densities is induced in the curved regions. The decrease in *E*CBM of the (13,0)@(21,0) DWBNNT is caused by the same mechanism as in the SWBNNTs mentioned above. Because the charge densities are distributed almost entirely in the innertube

 -




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**Figure 17.** Change in the CBM charge density of (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT during flattening deformation.

It should be noted that the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT show different changes in CBM charge densities from those of the SWBNNTs and (13,0)@(21,0) DWBNNT (Figure17). First, the CBM charge densities gradually transfer from boron atoms in the innermost tube to boron atoms in the second innermost tube. This charge delocalization and spreading account for the increase in *E*CBM in the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT during the early stage of the deformation. Then, overlap of the charge densities in curved regions in the second innermost tube is induced with increasing deformation. Consequently, *E*CBM of the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT decreases later in the deformation. It is this mechanism that results in the initial increase and subsequent decrease in *E*<sup>g</sup> in the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT. Because the latter has a larger *η*in than the former under the same *η* (see the atomic positions in the innermost tube at *η* of 0.43 in Figure 17), charge spreading from the first to the second innermost tube is completed earlier in the latter. Therefore, the latter shows an earlier shift from increase to decrease in *E*g.

force smallerthan that applied fortension ortorsion causes a larger decrease in the energy gap

Electronic Structure Calculations for Nano Materials 185

In Section 3, electronic structures of flattened MWBNNTs with the zigzag chiral index (*n*,0) were investigated by first‐principles DFT calculations. The key findings obtained in Section 3

**1.** When the chiral index of the innermost tube, *n*in, of a zigzag MWBNNT is larger than the critical one, *n*c, the energy gap decreases monotonically with increasing flattening

**2.** When *n*in < *n*c, the energy gap first increases and then decreases with increasing flattening compression. This initial increase and subsequent decrease are caused by a charge spreading from the first to the second innermost tube and by bond formation in curved

Department of Computational Science and Engineering, Graduate School of Engineering,

[1] Hohenberg P, Kohn W. Inhomogeneous Electron Gas. Physical Review 1964;136(3B)

[2] Kohn W, Sham LJ. Self‐Consistent Equations Including Exchange and Correlation Ef‐

[3] Jones RO, Gunnarsson O. The density functional formalism, its applications and

[4] Lundqvist S, March N. Theory of the Inhomogeneous Electron Gas. New York: Ple‐

[6] Kinoshita Y, Hase N, Ohno N. Flattening‐induced electronic changes in zigzag sin‐ gle‐ and multi‐walled boron nitride nanotubes: A first‐principles DFT study. Physical

[5] Martin R. M. Electronic Structure. Cambridge: Cambridge University Press; 2004.

and that the force required for flattening SWBNNTs is not unrealistic.

regions in the second innermost tube, respectively.

**3.** The *n*<sup>c</sup> is found to be an integer between 5 and 13.

Nobutada Ohno, Dai Okumura and Yusuke Kinoshita

fects. Physical Review 1965;140(4A) A1133‐A1138.

prospects. Reviews of Modern Physics 1989;61(3) 689‐746.

Nagoya University, Nagoya, Japan

are summarized as follows:

compression.

**Author details**

**References**

B864‐B871.

num Press; 1983.

Review B 2009;80(12) 125114.

### *3.2.3. Critical diameter of the innermost tube*

It is evident that a critical chiral index, (*n*c, 0), exists for flattened zigzag MWBNNTs. Consider the innermost tube of a zigzag MWBNNT, denoted as (*n*in, 0). In the case of *n*in > *n*c, *E*<sup>g</sup> decreases monotonically as the flattening deformation increases. In the reverse case (*n*in < *n*c), *E*<sup>g</sup> first increases and then decreases. From the results obtained in this study, *n*<sup>c</sup> is proven to be an integer between 5 and 13. Furthermore, if zigzag BNNTs have the same *n*in, a zigzag BNNT containing more walls shows a more rapid change in *E*<sup>g</sup> (compared with the change in *E*<sup>g</sup> between (13,0) SW and (13,0)@(21,0) DWBNNTs and between (5,0)@(13,0) DW and (5,0)@(13,0)@(21,0) TWBNNTs in Figure 13).

In the aforementionedexperimental study on a bentMWBNNT, a notable tendency for a zigzag atomic arrangement and local flattening have been observed [30]. Judging from the HRTEM images, *n*in of the MWBNNT is larger than *n*c. Therefore, it can be said that a possible reason for the change from insulating to semiconducting in the bent MWBNNT is electronic changes, as shown in Figure 16. To the best of our knowledge, there has been no experimental study on deformed BNNTs with *n*in< *n*c, but the results for the (5,0)@(13,0) DWBNNT and (5,0)@(13,0)@(21,0) TWBNNT obtained in this study are expected to be good predictions. Bending experiments on BNNTs with *n*in < *n*<sup>c</sup> are greatly anticipated.

### **4. Summary**

In Section 2, the electronic structures of (*n*,0) zigzag SWBNNTs subjected to tension, torsion, and flattening were investigated by first‐principles DFT calculations. The results revealed that the three deformation modes decrease the energy gaps of the SWBNNTs because of the decrease in the CBM energy caused by an overlap of CBM charge densities between circum‐ ferentially neighboring boron atoms. The key findings of Section 2 are that flattening with a force smallerthan that applied fortension ortorsion causes a larger decrease in the energy gap and that the force required for flattening SWBNNTs is not unrealistic.

In Section 3, electronic structures of flattened MWBNNTs with the zigzag chiral index (*n*,0) were investigated by first‐principles DFT calculations. The key findings obtained in Section 3 are summarized as follows:

