The Double Wall Boron Nitride Nanotube : Nano-Cylindrical Capacitor

Experimental results have shown that a small single wall carbon Nanotubes (SWCNTs) can be usually found inside “multi-walled” (MWCNTs). In this work it has been reported the stabilities and electronic structure of the single wall boron nitride Nano-tube (SWBNNTs) inside SWBNNTs. It has been shown that the gap energies and that energy of BNNTs are strongly dependent on their diameters or chirality. When this kind of BNNTs are inserted in the larger one of SWBNNTs, the gap energies of the double walled (DWBNNTs ) would even be much decreased due to the “coupled effect” of “wall buckling” difference and inter-wall p-p* “hybridization”. DWBNNTs are used for a theoretical study of a cylindrical molecular capacitor, including an inner cylinder with a positive charge distribution and an outer cylinder with a negative charge distribution. Due to their semiconductor characteristic and dielectric functionalities of SWBNNTs, DWBNNTs can be used as a cylindrical capacitor for the electronic devices. Keyword: single-wall BN-nanotubes, nano-cylindrical capacitors


INTRODUCTION
NSCCE or "Nanometer-scale capacitive charging effect" currently is familiar, e.g., the "coulomb blockade phenomenon" in the quantum dots 1,2 .However, they are arduous for resolving the detailed radial charges repartition within the nanostructures.Using SWBNNTs@SWBNNTs double wall tubes;it has been discussed the "radial" charge distributions of multi layered molecular capacitors.
Designing and control of Nano-tubes diameters are requirement to develop Nano-tube growth method.Nano-tubes with the diameters of less than one nanometer provide the ideal Nanospaces in X-dimension 3 .A few years ago, it has been suggested which the sizes of the catalysts used in the metal catalyze CVD (chemical vapor deposition) can explain the diameter of grown carbon nanotube 4 .Thisopinion has been confirmed by the seeing that a catalytic particle in the end of chemical vapor deposition grown nanotube has size proportional with the Nano-tubes diameters 5 .Chin Li Cheung, illustrate clearly the concepts of different sizes nano cluster catalysts which they can be used for controlling the structures and diameters of "CVD" grown nanotubes 6 .Smalldiameter carbon nanotubes indicate to many exotic properties such as an-isotropic optical absorption spectra 7 and super conductivities emanate from a "Pearle" distortion 8,9 .This finding has stimulated much fondness in studies of a small nanotube in both theoretically and experimentally [10][11][12][13][14] .BN-Nanotubes(BNNTs),which firstly synthesized and predicted through Chopra work and Rubiorespectively 15,16 , has a unique structural syllogism to carbon Nano-tubes but, contrary to the C-N-T being semiconductor or metallic depending on their chirality 16 .BNNT is usually can be used as an insulator regardless of its diameter and helicity or the number of walls [15][16][17] .
Experimental result has shown that a small SWNT is usually found inside a multi-walled CNT [11][12][13] .Therefore, there are strong incentive for studding in details the stabilities and interaction of narrow BNNT inside a bigger one in viewpoint of diameter, which makes easier for understanding the experimental result.Furthermore, the studies of double-walled boron nitride Nano-tubes (DBNNTs) have displayed interesting variations in its electronic properties comparing with those of free-standing part of BNNTs 18 .So it is also significant for seeing the interaction energies associated and inter wall coupling behavior with the narrow BNNTs 17,18 .
B-Nnanotube possessesgrate band gap around5 to 6 eV regardless of diameters, chirality, electronic properties and the number of walls15, 19.Furthermore, they are stable in view-point of mechanical and chemical structures 20 .Therefore, narrow single-wall BNNT can widely be applies as an ideal Nano-tube for the Nanosciences for producing suitable material such as; capacitor, atomic wire, and semiconductor 21, 22.In our study the SWBNNs are special material as an insulator for producing Nano-cylindrical capacitors 22 .Ryo Nakanishi reported an important synthesis method of a narrow SWBNNT having uniform distribution of diameter around 0.7+ 0.1 nm 1 .Their strategies to synthesize thin BNNTs are for combining the Nano-template reaction using Single wall carbon Nano-tubes which has developed in the past ten years 23 .In their systems, precursor molecules including Ammonia Borane Complexes or ABC, including boron and nitrogen were en-capsulated first in the single wall carbon tubes followed by suitable thermal decomposition-fusion 22 reaction inside the SWCNTs 22,23 .
It has been used arc-grownSWCNTs 24 with distribution (diameter)around1.4nm 24 as a model for synthesizing narrow SWBNNT 22,24 .Those distributions of SWCNT-models are essential for realizing the diameter selective synthesizing of BNNTs.
In this investigation, the systems have been simulated based on the various distribution diameters of SWBNNTs @SWBNNTs corresponding the experimental results of Ryo Nakanishiresults 1 .So we have started for answering to some questions for the mechanism of the radial charge distributions on the inner and outer electrodes, band gapes, potential difference 23 between two layers of the Nano cylindrical 24 capacitor and the capacitance of our system when the inner tubes are semiconducting 25 , and the others are metallic 26 .
Gugang Chen in the studies of doped double-walled carbon nanotubes 26 exhibited the Resonant-Raman-Scattering (RRS) from the phonons 26 on each carbon shell determines the radial charges distributions 26 .The self-consistent including tight-binding model (SCTB) conûrms 25 the observed molecular faraday-cage effects, so most of the charges reside on the outer-walls, even when these walls were originally 26 semiconducting and the inner-walls were metallic [24][25][26] .
Those systems have been modeled as three-layer cylindrical capacitor within bromine anion forming the shell (around the outer nanotubes 26 ).The total energies contain3terms including the innertube, outer tube and band structures of the electrostatic energies E es for the three-layer charge distributions: The signs"i" and "k" label the wave vector and occupied band for the outer or inner tube 26 .
They assumed that the surpluses charges on both shells are distributed in the innitely thin-walls at the nuclear radius of those shell 26,27 .The resulting electrostatic energies of the triple-walled capacitors are: where {ε o } is the permittivity of free space 27 , L is the unit-cell length of the outer tubes and (n) is the linear densities of surpluses holes for the innertubes.
There are two physical properties acts in concert 27 for isolating most of the holes into the outer nanotubes.(1):the gap band of the thinner diameter tubes towards the larger one, so they empty last 28 .(2) The cylindrical-geometry rather raises the electrostatics potentials in the inner tubes 27 .Lonely the charges on the inner tubes in the chirality of Zigzag (n,0) of BNNTs are anticipated to have direct band gaps 27 .On the other side the armchair (n, n) of BNNTs will have indirect band gaps 27 .Because of its large band gap around 5 eV, experiments 28 using BNNT as the conduction-channel 29 for field effect transistors 29 or FETs showed that BNNT allowed transport through only the valence band 28 .The other important features about the band gaps of BNNTs are that they are tunable by doping with carbons 29 , radial deformation 30 , or by applying the transverses electric fields through the BNNTs so-called giantstark-effect [31][32][33] .
Theoretical band structure calculation suggested that "SWBNNT" can either be n-type or p-type semiconductor by controlling the composition of carbon into "SWBNNTs".Carbon impurities on a boron site result in electron carriers while on a nitrogen site result in holecarriers 34 .
In this study it has been exhibited that the piezo electricity 34 for SWBNNTs causes for increasing the capacities of SWBNNTs@SWBNNTs capacitor comparing to SWCNTs@SWCNTs.This phenomenon 35 originated from the deformation effects due to the tumbling of the planar hexagonal boron nitride network to produce tubular structures 35 .It has been exhibited by Nakhmans on that BNNT could be excellent piezoelectric systems 36 .As instant, piezoelectric constants for variant zigzags of SWBNNT were found for increasing along with the decreasing of the radius in several BNNTs 36 .Experimentally 37 , Bai has exhabited that under in situ elastic bending deformation 36 or EBD at room temperature high-resolution-transmission-electronmicroscope 36,37 , a normally electrically insulating 37 MWBNNT may transform to a semiconductor 37 .K. Uchida et al., [38][39] in a discussion of quantum effect in the cylindrical carbon nanotubes capacitor exhibited that the distributions of the accumulated charges in the inner tubes are quantum mechanically 37 spilled outward, while that in the outer tubes are penetrating inwards 38 .They have shown the reflecting those charges spills, the electrostatic capacitance of the systems are larger than what would be expected from the classical theories 38 .
Finally, they have shown that the capacitance exhibit two principal quantum effects, (1): the capacitance shows a large bias dependence 38 , reflecting the densities of states of the carbon Nano tube electrodes.( 2): the capacitances are enhanced according to a quantum mechanical spill of the stored electrons density from the tubes walls of the CNTs [36][37][38] .
Based on our previous works  , we simulated our model in viewpoint of different band gap energies via considering the single wall boron nitrides as both inner and outer tubes with variant diameters and chirality in the ranges of (6.0 <d<8.0А) and (11.0<d< 16.0А) for inner and outer tubes respectively.

Theoretical background
Carbon nanotube is thin seam less graphitic cylinders 39 , which exhibit an unusual combination of the Nano-meter size diameter and Milli-meter size length 39 .These topologies, are included with the absence of defects 38 on the macroscopic scales 62 , yields to uncommon electronic properties of individual 62,63 SWCNTsthat depends on their diameters and chirality 63 , can be either 63 insulating, metallic or semiconducting 64,65 .
Consider a cylindrical capacitor of length "d l ", inner radius "R inn ", outer radius "R out ", and with charges Q=d l .q l which q l is the charges per unit length (magnitude) on each cylinders (Scheme1).Assume "d l ">> "R inn " or "R out " and Neglect fringing and electric elds between cylinders: use Gauss' law E(2prd l ) = d l .ql /ε o = E(r) = q l / 2pε o r (1) and electric potentials between cylinders: use V out =0 V(r) =- (2) And V = V + _ V = V (inn) -V (out) = Q/2p ε oL ln R out / R inn (3) and capacitances for cylindrical geometries are: (4) which "K" is the dielectric constant of the system 65 .∆ V (inn-out) , is positive quantities because 2kln (R inn /R out ) is a positive quantities these because outer layers are at a higher potential than the inner layers.

Computational details
Calculations were accomplished using GAMESS-US packages 72 .In this work, it has been mainly focused for optimization of each tube with DFT methods [73][74][75][76][77] consist of the m06 and m06-L 74 .The m062x 74 , m06-L 74 , and m06-HF 74 are a unique Meta hybrid 74 DFT functional with a good correspondence in non-bonded 78 calculations and are useful for calculating the energies of the distances between two coaxial cylinders of radii R inn R out and in the cylindrical capacitor 73 .The Perdew, Burke and Ernzerh of (PBE) 75 exchange correlation 75 (XC) functional 74,75 of the generalized-gradient-approximations 95 (GGA) are adopted.The lattice constant has been optimized for the atomic coordinate and has done through the minimization of the total energies.For geometries optimizations, all the internal coordinates were relaxed until the Hellmann-Feynman-forces 74 was less than 0.005 angstrom.
At each inter tube configurations, a single point calculation is carried out and the total energies are recorded.The resulting sliding rotation energy surface isused for fixing our model in a better position.
We employed DFT theories with the van der Waals DFT for modeling the exchange-correlation energies of SBNNTs and SWCNs 76 .The {ξ-basis set} with polarization 76 orbital was used for single wall tube 76 .
For non-covalent approaches, DFT methods disable for describing van der Waals 73,77.The other functional are correctly insufficient for showing the correlation and exchange energies in non-bonded medium-ranges distances.Furthermore, recent study has illustrated that the medium-range exchange 78,79 Scheme 1 In our systems the calculation of the nano-cylindrical capacitors can be obtained from the electrical potential in the spaces between two coaxial cylinders of radii R inn and R out and finite length d l in the z direction, 0 ≤z≤ d l .We supposed the geometrical capacitances in our systems are a function of d l , R inn R out or C g = F(d l , R inn R out ). Bilalbegovi [66][67][68] with a molecular dynamic simulation and extension series based on Bessel functions 69 has modified and discussed that capacitance around the shorter coaxial cylinders of radios 69 .So the capacitance in our model can be calculated via (5) in the Finite Nano-meter scales cylindrical-capacitor based on the classical electrodynamic 85 .For calculation the capacitance for the eq.4 and eq.5 for the potential difference applied between two cylindrical plates V= V (inn) -V (out) has been calculated by pop = chelp G commands 76 .energies leads to the large systematic errors 78 in the prediction of molecular propertie 79 .

SWBNNTs
We further calculated the interaction energies between two coaxial cylinders of radii "R inn " and "R out " for SBNNTs and SWCNs in the structures.The dielectric permittivities as function of dielectric size were determined via Abinito calculation [78][79] .The interaction energies for capacitor were calculated via an extended huckel method in all items according to the eq.6.∆E s (eV) = {E total -(2E SWNT + E SWBNNT )} + E BSSE (6) Where the "∆E s " is the stability energy of capacitor.
The MESPs are calculated and distributed at large number of grid points 84 in a cube regularly.
The representative 85 atomic charges 86 would be computed as averages values over a few molecular conformations for the molecules 86,87 .

RESULT AND DISSCUSSION RESULT AND DISCUSSION
We first consider the h-BN sheet and 3D BN tubes in various diameters and chirality where the (5, 5) @ (8, 8) structure is found as a stable form compared to other forms.The stability depends on the distance between inner radius "R inn ", and outer radius "R out in one hand and the chirality on other hand Table1.The minimum energies are calculated based on eq.9 in terms of the total energy of the optimized structures and are listed in Table1.
The differences in the band structure and Fermi 88 level energy of different tubes have been calculated.Furthermore, we have presented the number of states in unit energy interval through density of states (DOS) (Fig. 1).DOS 88 was plotted as a curved map and we have considered those graphs as a tool for analyzing the nature of electronic structure in our systems 88,89 .The original total DOS (TDOS) 89 of our system was calculated based on References {108-110} 108-110 in Figs. 2, 3, 4and 5.
The interaction energies for capacitors were calculated in all items according to the eq.6 ∆E s (eV) = {E c -(2E Dopant -G + E n -BN )} + E BSSE Where the "∆E s " is the stability energy of capacitor.
The wall buckling has been defined as the differences between the mean radiuses 91 of the cylinders consisting of the B and N atoms.It has been shown that the buckling rapidly increases 92 with decreasing of the tube diameter 91 and are independent of the chirality 92,93 .
The (5,5) SWBNNTs in the form of inner cylindrical are semiconductors due to the existence of an energy gap in the range of (7.55-9.67)KJ/mol (Table1) which are between the valence band and the conductor 93 band.Those armchair tubes have a direct band gap, similar to the BN structure 94 .
It is indicated in this study that the DWBNNTs can be used as a capacitor due to the semiconductor character, metallic behavior and dielectric function of SWBNNTs.Surprisingly, the optical properties 93 of the SWBNNTs optimized by Guo 94 found that the dielectric function 93 could also be divided into two spectral regions 94 , namely, the high and the low energy region -energies these systems 94 .
We show that for the (5, 5) @ (8, 8) the distribution of the accumulated charges in the inner tubes are quantum-mechanically spilled outwards clearly.The capacitance ( C q and C g ) and dielectric constant of the various capacitors have been calculated based eq.4 and are listed in table.2.

CONCLUSION
This work has been investigated the electric polarization and capacitance in a nano-scale coaxial-cylindrical-capacitors which is made of DW boron nitride, (n, n) @ (m, m), using ONIOM model including density-functional and semi empirical methods with the enforced Fermi-energies difference scheme.Despite the fact that the SWBNNTs@ SWBNNTs are a cylindrical capacitor, it has a very sensitive electrical storage compared to SWCNTs@ SWCNTs.

LUMO/HOMO Fermi energy l n R out /R inn
| Σ Q | ∆E