Appl Phys A (2012) 108:283–289DOI 10.1007/s00339-012-6933-3

Carbon dioxide detection by boron nitride nanotubes

Hamze Mousavi · Jamshid Moradi Kurdestany ·Mehran Bagheri

Received: 10 February 2012 / Accepted: 10 April 2012 / Published online: 27 April 2012© Springer-Verlag 2012

Abstract The effect of gas molecule adsorption is inves-tigated on the density of states of (9,0) zigzag boron nitridenanotube within a random tight-binding Hamiltonian model.The Green function approach and coherent potential approx-imation have been implemented. The results show that theadsorption of carbon dioxide gas molecules by boron atomsonly leads to a donor type semiconductor while the ad-sorption by nitrogen atoms only leads to an acceptor. Sincethe gas molecules are adsorbed by both boron and nitrogenatoms, a reduction of the band gap is found. In all cases,increasing the gas concentration causes an increase in theheight of the peaks in the band gap. This is due to an in-creasing charge carrier concentration induced by adsorbedgas molecules.

1 Introduction

Theoretical studies of boron nitride nanotubes (BNNTs),have led to a renewal of interest in experimental studies

H. Mousavi (�)Department of Physics, Razi University, Kermanshah, Irane-mail: hamze.mousavi@gmail.comFax: +98-831-4274556

H. MousaviNano Science and Nano Technology Research Center, RaziUniversity, Kermanshah, Iran

J.M. KurdestanyDepartment of Physics, Indian Institute of Science, Bangalore560 012, India

M. BagheriCondensed Matter Group, Laser and Plasma Research Institute,Shahid Beheshti University, G.C., Evin, Tehran 19835-63113,Iran

[1–3]. Since the BNNTs’s structure in hexagonal form isanalogous to carbon nanotubes (CNTs), boron (B) and nitro-gen (N ) atoms can be replaced by the carbon atoms (Fig. 1).Although CNTs and BNNTs have similar structures, buttheir properties are quite different. Theoretical studies sug-gest that BNNTs are wide gap semiconductors with a uni-form electronic band gap, Eg ∼ 5.5 eV [1, 4], and this bandgap is independent of either the diameter or chirality of BN-NTs (Fig. 2) [1, 4–9] while metallic or semiconducting be-havior in CNTs is observed from the electrical conductance[10].

Gas sensors are widely used in various fields, e.g., inindustry for environmental analysis, in medical diagnosticsetc. However, CNTs have the potential to be developed as agas sensing material due to their inherent properties such astheir small size, great strength, high electrical, and thermalconductivity, and high specific surface area [10–16]. Simi-lar to CNTs, the effects of gas adsorption on the electronicproperties of BNNTs have attracted certain attention [17–23]. For example, the adsorption of molecular hydrogen onBNNTs has been studied by using density functional the-ory (DFT) [17, 18]. They found an even–odd oscillation be-havior of the adsorption energy of hydrogen atoms on thetube. Using molecular-dynamics simulations, collision andadsorption investigated by Han et al. [19]. They discussedthat at incident energies below 14 eV hydrogen bounces offthe BNNT wall and at incident energies between 14 and22 eV each hydrogen molecule is dissociated at the exteriorwall. Using DFT within the generalized gradient approxima-tion, methane adsorption onto BNNT and CNT is consideredby Ganji et al. [20]. They showed that the methane moleculeis preferentially adsorbed onto the CNT with a binding en-ergy of −2.84 kcal/mol. They also found that the methaneadsorptive capability for the exterior surface increases forwider CNTs and decreases for wider BNNTs. Adsorption of

284 H. Mousavi et al.

Fig. 1 Geometry of hexagonal BN sheet. Each Bravais lattice unitcell includes two nonequivalent sites which are denoted by B1 and N1.Three vectors r01, r02 and r03 connect a B1 (N1) site to its nearest-neighbor N1 (B1) sites. The primitive vectors are (a1,a2) and a0 is

interatomic distance. Zigzag BNNTs can be thought of as a tube rolledfrom a hexagonal sheet of BN in x direction. The figure also shows aschematic adsorption of two molecules by B1 and N1 sites

Fig. 2 The DOS of pure zigzag (9,0), armchair (5,5) BNNTs and alsohexagonal sheet of BN. The figure indicates that this structures arewide gap semiconductors with a uniform electronic band gap which isindependent of system’s geometry

carbon dioxide (CO2) on the BNNT is one of the interest-ing cases for detection of gases by nanomaterials [24–28],for example, using ab initio calculations, plane waves andlocalized atomic orbitals; strong CO2 adsorption on boron-rich BNNT has been recently reported [24].

Before we present the details of our study, we summa-rize our principal results. We present an extensive study ofthe effect of CO2 (in general form it is denoted by XY2) ad-sorption on density of states (DOS) of (9,0) zigzag BNNTin the random tight-binding model and for this we developthe Green function method. Since the gas molecules are ad-sorbed randomly by the BNNT’s atoms, we need to cal-culate the configurationally averaged Green function [29,30] by using the coherent potential approximation (CPA).The remaining part of this paper is organized as follows. InSect. 2, we introduce the model and obtain the equation ofmotion within the random tight-binding Hamiltonian model.

In Sect. 3, the CPA method is introduced. In the two lastsections, we present the numerical results, discussion andconclusion.

2 Model and framework

We study the random tight-binding model defined by theHamiltonian

H = −∑

ijαβ

[t0αβij − (

εα0i + εα

i

)δij δαβ

]cα†i c

βj , (1)

where α and β refer to the B1 or N1 sites inside the Bravaislattice unit cell (Fig. 1), t

0αβij is the amplitude for a π elec-

trons to hop from site α in the unit cell i to the site β in thenearest-neighbor unit cell j; εα

0i = ε0 (−ε0) is the on-site en-ergy of B (N ) atom in the Bravais lattice unit cell i; εα

i is therandom on-site energy for sub-site α in the Bravais latticeunit cell i in which it takes a value of zero with probability1 − c for host (B1 or N1) sites and nonzero with probabilityc for impurity (gas adsorbed) sites, where c is concentra-tion of gas molecule adsorption and c

α†i (cα

i ) is the electroncreation (annihilation) operator at the site α in the Bravaislattice unit cell i. In our calculations we take the chemicalpotential μ = 0, which corresponds to contribution of oneelectron per pz orbital in the system. We assume the unitswith � = 1.

Since each B (N ) atom in the Bravais lattice unit cell canadsorb a XY2 gas molecules (Fig. 1), the Hamiltonian of thesystem is given by a 8 × 8 matrix with the following basiskets of the Hilbert space:{∣∣ΦB1

⟩,∣∣ΦB2

⟩,∣∣ΦB3

⟩,∣∣ΦB4

⟩,∣∣ΦN1

⟩,∣∣ΦN2

⟩,∣∣ΦN3

⟩,∣∣ΦN4

⟩},

(2)

where B1 (N1) denotes B (N ) atom in the Bravais latticeunit cell while B2 (N2), B3 (N3) and B4 (N4) are adsorption

Carbon dioxide detection by boron nitride nanotubes 285

atoms XY2 (Fig. 1). So, the Green function is given by a8 × 8 matrix,

G(i, j ; τ) =(

GBB(i, j ; τ) GBN(i, j ; τ)

GNB(i, j ; τ) GNN(i, j ; τ)

), (3)

in which τ denotes the imaginary time and for a typical 4×4sub-matrix, GBB(i, j ; τ) can be written as

GBB(i, j ; τ) =

⎛

⎜⎜⎝

GB1B1(i, j ; τ) GB1B2(i, j ; τ)

GB2B1(i, j ; τ) GB2B2(i, j ; τ)

GB3B1(i, j ; τ) GB3B2(i, j ; τ)

GB4B1(i, j ; τ) GB4B2(i, j ; τ)

GB1B3(i, j ; τ) GB1B4(i, j ; τ)

GB2B3(i, j ; τ) GB2B4(i, j ; τ)

GB3B3(i, j ; τ) GB3B4(i, j ; τ)

GB4B3(i, j ; τ) GB4B4(i, j ; τ)

⎞

⎟⎟⎠ , (4)

where Gαβ(i, j ; τ) = −〈T cαi (τ )c

β†j (0)〉, and T is the time

ordering operator.Using Hamiltonian in Eq. (1) and the Green function

technique, the equation of motion for the electrons can bewritten as

∑

[(−I

∂

∂τ+ ε0i + εi

)δi + t0

i

]G(, j ; τ) = δ(τ )δij I,

(5)

where I is a 8 × 8 unit matrix and δ(τ ) is a Dirac δ-function.Using an imaginary time Fourier transformation,

G(, j ; τ) = 1

β

∑

n

e−iωnτ G(, j ; iωn), (6)

and the following relation [31]:

1

β

∫ β

0dτei(ωm−ωn)τ = δmn, (7)

we obtain∑

[(iωnI + ε0i + εi )δi + t0

i

]G(, j ; iωn) = δij I, (8)

where β is the inverse of temperature, ωn = π(2n+1)/β arethe fermionic Matsubara frequencies and {m,n} are integernumbers. Analytical continuation, iωn → E = E + i0+, ofEq. (8) leads to the following equation:∑

[(EI + ε0i + εi )δi + t0

i

]G(, j ;E) = Iδij . (9)

Since in Eq. (9), the random Green function matrixG(i, j ;E), could not be calculated exactly, it should be ex-panded in terms of the pure system’s Green function matrix,G0(i, j ;E), and the random potential [32],

G(i, j ;E) = G0(i, j ;E) +∑

′G0(i, ;E)V′G

(′, j ;E)

,

(10)

where G0(i, j ;E) is given by

G0(i, j ;E) = 1

N∑

k

eik.rij (EI + ε0 − εk)−1, (11)

N is number of Bravais lattice unit cells in the system, εk

is the Fourier transformation of t0i and rij ’s are three vec-

tors that connect a B1 (N1) site to its nearest-neighbor sites(Fig. 1),

r01 = a0ey, r02 = a0

2

(√3ex − ey

),

(12)r03 = −a0

2

(√3ex + ey

).

Here a0 is interatomic distance, {ex, ey} are the unit vectorsin BN plane, k = kxex + kyey is a two-dimensional wavevector in the first Brillouin zone with the following compo-nents:

kx ∈[−4

3

(π

a

),+4

3

(π

a

)],

(13)

ky ∈[− 2√

3

(π

a

),+ 2√

3

(π

a

)],

where a = |a1| = |a2| = √3a0, and {a1,a2} are primitive

vectors (Fig. 1) [33],

a1 = aex, a2 = a

2

(−ex + √3ey

). (14)

We note that the periodic boundary condition in the x di-rection implies that exp(ikxL) = 1, where L = 2πR andR is the BNNT’s radius and kx is restricted to kx = 2πr

L=

2πr

a√

p2+pq+q2, where {p,q, r} are integers. Also, the nonzero

elements of the hopping matrix, t0ij , can be written as,

t0B1N1〈ij〉 = t

0N1B1〈ij〉 ≡ t (here 〈ij 〉 denotes the nearest-neighbor

sites), and therefore the Fourier transformation of t0ij can be

represented by

εk =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 εk 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0ε∗

k 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (15)

in which εk can be obtained,

εk = t

[2 cos

(kxa

2

)exp

(−i

kya

2√

3

)+ exp

(ikya√

3

)]. (16)

Here εk is radius dependent due to kx = 2πrL

= rR

. Using Eq.(11), we can obtain the DOS of the pure system (Fig. 2),

D0(E ) = − 1

2πIm

[G0B1B1(i, i;E) + G0N1N1(i, i;E)

]. (17)

286 H. Mousavi et al.

We investigate the DOS of (9,0) BNNT, when gas moleculesare adsorbed on the system. In Eq. (10), the random poten-tial matrix, V′ , is defined by V′ = εδ′ + δt′ , whereδt′ = t′ − t0

′ is hopping integral deviation matrix withrespect to the pure system [34]. In case of adsorption of theXY2 gas molecules by the B1 sub-site in the Bravais latticeunit cell, if the hopping terms to the nearest neighbors in therandom potential matrix are allowed, then the matrix can berepresented by

V′ =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 δB1 δB

2 δB2 0 0 0 0

δB1 εB

1 δB3 δB

3 0 0 0 0

δB2 δB

3 εB2 δB

4 0 0 0 0

δB2 δB

3 δB4 εB

2 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (18)

and if the molecules are adsorbed by N1 sub-site, V′ canbe written as

V′ =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 δN

1 δN2 δN

2

0 0 0 0 δN1 εN

1 δN3 δN

3

0 0 0 0 δN2 δN

3 εN2 δN

4

0 0 0 0 δN2 δN

3 δN4 εN

2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (19)

and if the molecules are adsorbed by both B1 and N1 sub-sites, the random potential matrix can be written by

V′ =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 δB1 δB

2 δB2 0 0 0 0

δB1 εB

1 δB3 δB

3 0 0 0 0

δB2 δB

3 εB2 δB

4 0 0 0 0

δB2 δB

3 δB4 εB

2 0 0 0 0

0 0 0 0 0 δN1 δN

2 δN2

0 0 0 0 δN1 εN

1 δN3 δN

3

0 0 0 0 δN2 δN

3 εN2 δN

4

0 0 0 0 δN2 δN

3 δN4 εN

2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (20)

where δαs ’s (s = 1,2,3,4) are schematically indicated in

Fig. 3. Also, εB1 (εN

1 ) differs from on-site energies of hostB (N ) atom and adsorbed gas X atom and εB

2 (εN2 ) differs

from on-site energies of host B (N ) atom and Y atoms.The Dyson equation for the average Green function cor-

responding to Eq. (10) can be written as [35]

G(i, j ;E) = G0(i, j ;E) +∑

′G0(i, ;E)

× Σ(, ′;E)

G(′, j ;E)

, (21)

Fig. 3 A schematic presentation of the nonzero elements of hoppingintegral deviation matrix (Eqs. (18)–(20))

where the self-energy, Σ(, ′;E), is defined by∑

′Σ

(, ′;E)

G(′, j ;E) =

∑

′

⟨V′G

(′, j ;E)⟩

. (22)

Here, 〈· · ·〉 denotes configurational average. The Fouriertransformation of Eq. (21) is as follows:

G(i, j ;E) = 1

N∑

k

eik·rij{[

G0(k;E)]−1 − Σ(k;E)

}−1,

(23)

where Σ(k;E) is the Fourier transformation of the self-energy, Σ(, ′;E), and G0(k;E) can be written by

G0(k;E) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

E+ 0 0 0 εk 0 0 00 E 0 0 0 0 0 00 0 E 0 0 0 0 00 0 0 E 0 0 0 0ε∗

k 0 0 0 E− 0 0 00 0 0 0 0 E 0 00 0 0 0 0 0 E 00 0 0 0 0 0 0 E

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

−1

,

(24)

where E± ≡ E ± ε0. In general there is no analytical solu-tion for such random systems; hence they should be solvedapproximately. We use the CPA to obtain a route for the dis-ordered systems.

3 Coherent potential approximation

Since the gas molecules are adsorbed randomly by theBNNT’s atoms, the Green function in the equation of mo-tion is random and the local behavior could be different from

Carbon dioxide detection by boron nitride nanotubes 287

total system behavior. Hence we should calculate configura-tional average properties. We treat this in the CPA methodto take the average over all possible adsorbed molecule con-figurations. In the CPA, inter-site correlations are neglectedand each lattice site is replaced by an effective site exceptone, which is called the impurity (molecule adsorbed) siteand is denoted by i. Then the self-energy is local and takesthe same value for all sites, Σ(i, j ;E) = Σ(E)δij , so Eqs.(22) and (23) at impurity site reduce to the following:

Σ(E) = ⟨ViiGimp(i, i;E)

⟩G−1(i, i;E), (25)

G(i, i;E) = 1

N∑

k

{[G0(k;E)

]−1 − Σ(E)}−1

. (26)

We point that in Eq. (26), the self-energy is independenton k. Equation (26) in matrix form is as follows:

G(i, i;E) = 1

N∑

k

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ξB1B1+ 0 0 00 ξB2B2 0 00 0 ξB3B3 00 0 0 ξB4B4

ε∗k 0 0 00 0 0 00 0 0 00 0 0 0

εk 0 0 00 0 0 00 0 0 00 0 0 0

ξN1N1− 0 0 00 ξN2N2 0 00 0 ξN3N3 00 0 0 ξN4N4

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

−1

,

(27)

where ξαα± ≡ E ± ε0 − Σαα(E) and ξαα ≡ E − Σαα(E).Using Eqs. (10) and (21), the impurity Green function,Gimp(i, i;E), relates to the average Green function,G(i, i;E), by

Gimp(i, i;E) = G(i, i;E) + G(i, i;E)

× [Vii − Σ(E)

]Gimp(i, i;E). (28)

The new average Green function, Gnew(i, i;E), is obtainedby taking average over all possible adsorbed molecule siteconfigurations,

Gnew(i, i;E) = ⟨Gimp(i, i;E)

⟩. (29)

To calculate 〈Gimp(i, i;E)〉, we point that since site i in-cludes two non equivalent sub-sites B1 and N1, then fourpossible configurations at this site are B1 − N1 (both sub-sites B1 and N1 of site i are pure), B1 − XY2 (sub-site B1 ispure while XY2 is adsorbed on sub-site N1), XY2 −N1 (XY2

is adsorbed on sub-site B1 while sub-site N1 is pure) andXY2 − XY2 (XY2 is adsorbed on both sub-sites B1 and N1

of site i). The probability of these configurations are P1 =

Fig. 4 The DOS of (9,0) BNNT in the pure case and when CO2 gasmolecules are adsorbed by B atoms. Concentrations of gas moleculesare chosen to be c = 0.0 and c = 0.040 in panel (a) and c = 0.005,c = 0.010, c = 0.020 and c = 0.040 in panel (b). In panel (a), densitiesof states of pure and random system are compared in full range ofenergy. In panel (b), D(E) is plotted in gap energy range. For a valueof gas concentration, there is a peak in the conduction band because ofadsorption of XY2 by B atoms. The height of the peak increases whenconcentration of gas increases

(1 − c)2, P2 = (1 − c)c, P3 = c(1 − c) and P4 = c2,respectively. Equations (24)–(29) should be solved self-consistently to provide the average Green function, G(i, i;E),in the CPA formalism. So, we can obtain the DOS of disor-dered system by

D(E ) = − 1

2πIm

[GB1B1(i, i;E) + GN1N1(i, i;E)

]. (30)

4 Numerical results and discussion

As we noted above, the gas molecules are adsorbed ran-domly by BNNT’s atoms, so the Green function should becalculated using the configurational average method. By cal-culation of the average Green function in the CPA formal-ism based on the random tight-binding model, the effect offinite CO2 gas molecule adsorption on the DOS of (9,0)BNNT is studied. First, we set the B and N on-site ener-gies as, ε0 � 0.8t [36–38] where t � 2.9 eV [36, 37]. Sec-

288 H. Mousavi et al.

Fig. 5 The DOS of (9,0) BNNT in the pure case and when moleculesare adsorbed by N atoms. Concentrations of gas molecules are chosento be c = 0.0 and c = 0.040 in panel (a) and c = 0.005, c = 0.010,c = 0.020 and c = 0.040 in panel (b). In panel (a), DOS of pure andrandom system are compared in full range of energy. In panel (b), D(E)

is plotted in gap energy range. For a value of gas concentration, thereis a peak in the valence band due to adsorption of CO2 by N atoms.The height of the peak increases when concentration of gas increases

ondly, we point out that if gas molecules are randomly ad-sorbed by B atoms, we set on-site energies and hopping in-tegral deviations as εB

1 = +0.8t , εB2 = +0.5t , δB

1 = +0.5t ,δB

2 = +0.3t , δB3 = −0.5t and δB

4 = −0.3t where XY2 leadssystem to charge donors. If gas molecules are adsorbed byN atoms, we can set the on-site energies and hopping in-tegral deviations, in the random potential (Eqs. (18)–(20)),as εN

1 = −0.8t , εN2 = −0.5t , δN

1 = −0.5t , δN2 = −0.3t ,

δN3 = +0.5t and δN

4 = +0.3t so that XY2 leads the systemto become a charge acceptor. Then using these parameterswe investigate the effect of finite gas molecule adsorptionon the system. As the DOS is sensitive to the adsorptionof these gases, for small concentrations of gas adsorptionc = 0.005, c = 0.010, c = 0.020 and c = 0.040, we foundthat the effect of gas adsorption causes a finite DOS at zeroenergy in contrast to the pure case, i.e., c = 0.0 where theDOS vanishes. Now, we investigate three cases. First we as-sume that the gas molecules are adsorbed by just B atoms.Figure 4 shows that there is a peak in conduction band fora value of gas concentration. So, due to adsorption of CO2

Fig. 6 The DOS of (9,0) BNNT in the pure case and when moleculesare adsorbed by both B and N atoms. Concentrations of gas moleculesare chosen to be c = 0.0 and c = 0.040 in panel (a) and c = 0.005,c = 0.010, c = 0.020 and c = 0.040 in panel (b). In panel (a), DOS ofpure and random system are compared in full range of energy. In panel(b), D(E) is plotted in gap energy range. There are two peaks in theconduction and valence bands due to adsorption of XY2 by B and N

atoms, respectively. The height of the peak increases when concentra-tion of CO2 increases

molecules, a donor (n) type semiconductor is obtained. Theheight of the peak in the band gap increases with increasein c because of the increase in charge carrier concentrationinduced by adsorbed gas molecules. Second, we consideranother case where the gas molecules are adsorbed by N

atoms. In Fig. 5 we show adsorption of small concentrationof CO2 causes a finite DOS in valence band and an accep-tor (p) type semiconductor is resulted. Finally, in Fig. 6 weshow a case of gas adsorption where molecules of gas arerandomly adsorbed by both B and N atoms. In this generalcase, we find that there are two peaks in the band gap fora value of c. Hence the band gap of system reduces due toadsorption of CO2 molecules.

5 Summary and conclusion

We have investigated the effect of adsorption of CO2 gasmolecules on the DOS of (9,0) BNNT. We found that, for

Carbon dioxide detection by boron nitride nanotubes 289

three different cases of adsorption, where CO2 moleculesare separately adsorbed by B , N and both B and N atoms,DOS of the system is sensitive to the adsorption of this gas.An n-type (p-type) semiconductor is resulted because of ad-sorption of gas molecules by B (N ) atoms. A reduction inthe band gap is calculated when molecules are adsorbed byboth B and N atoms. These are due to an increase in chargecarrier concentration induced by adsorbed gas molecules.So, it is shown that BNNT has the capability to detect CO2

gas. Therefore, it can be concluded that the study of the gassensing characteristics carried out in this work have shownBNNTs to have the potential to be an excellent CO2 sensormaterial.

Acknowledgement One of us (Jamshid MK) would like to thankPranab Jyoti Bhuyan for his helpful comments.

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