Boron Nanotube Structure Explored by Evolutionary Computations

Abstract

In this work, we explore the structure of single-wall boron nanotubes with large diameters (about 21 Å) and a broad range of surface densities of atoms. The computations are done using an evolutionary approach combined with a nearest-neighbors model Hamiltonian. For the most stable nanotubes, the number of 5-coordinated boron atoms is about $63\%$ of the total number of atoms forming the nanotubes, whereas about $11\%$ are boron vacancies. For hole densities smaller than about 0.22, the boron nanotubes exhibit randomly distributed hexagonal holes and are more stable than a flat stripe structure and a quasi-flat B₃₆ cluster. For larger hole densities (>0.22), the boron nanotubes resemble porous tubular structures with hole sizes that depend on the surface densities of boron atoms.

1. Introduction

Boron nanotubes (BNTs) have been synthesized for the first time in 2004 [1]. A magnesium-substituted mesoporous silica template (Mg-MCM-41) was applied by Ciuparu et al. [1] to prepare pure boron single-wall nanotubes at 870 °C with uniform diameters ($36\pm 1$ Å) using the mixture of ${\mathrm{BCl}}_3$ and ${\mathrm{H}}_2$ as gas sources. The authors attributed the Raman peaks at $210{\mathrm{cm}}^{-1}$ and between 300 and $500{\mathrm{cm}}^{-1}$ to typical tubular structures, where the first one ($210{\mathrm{cm}}^{-1}$) corresponds to the characteristic radial breathing mode. In 2010, Liu et al. [2] reported the first large-scale fabrication of single crystalline multilayered BNTs using boron $(99.99\%)$ and boron oxide powders $(99.99\%)$ as source materials. The as-synthesized BNTs had lengths of several micrometers and diameters in a range from 10 to 40 $\mathrm{nm}$. The nanotubes were cataloged as multilayered single crystalline BNTs with an interlayer spacing of about $3.2$ Å. Moreover, these nanotubes were experimentally proven to have metallic properties regardless of their chirality. The metallic behavior of BNTs makes them attractive in the design of novel electronic nanodevices such as field-effect transistors, light-emitting diodes, and field-emission displays, or for photosensitive device applications [3,4].

The experimental studies were preceded by several theoretical investigations, most often based on density functional theory (DFT). Quasiplanar [5,6], tubular [7,8], and convex and spherical [9] boron clusters have been computationally explored. Moreover, the existence of quasiplanar boron clusters [10,11] implies that boron fullerenes [12,13] and caped BNTs [14,15] may exist because larger quasiplanar or planar clusters will tend to remove dangling edge bonds by forming closed tubular or polyhedral structures.

Although some reports affirm that BNTs with diameters smaller than 1.7 nm [16,17] or 2 nm [18] show semiconducting behavior due to the band opening at the Fermi level through curvature-induced out-of-plane buckling of certain atoms, later calculations based on second-order Møller–Plesset perturbation theory [19] and dispersion-corrected DFT calculations [20] showed that the surface buckling is more likely an artifact of standard DFT approaches.

Thus, BNTs are found to be metallic, and this property is independent of the diameter and chirality (armchair or zigzag) of the nanotube [21]. Moreover, there have been a lot of morphologies of BNTs in contrast to only one morphology of carbon nanotubes (CNTs) [22]. A more detailed description of the structure and properties of 1D-boron structures can be found in Refs. [4,23].

Crystal structure prediction (CSP) is a long-standing challenge in physical and materials sciences. Several methods have been applied for materials design [24,25]. Evolutionary algorithms, such as genetic algorithms (GAs) [25] that use human evolution mechanisms (such as crossover and mutation), have been used, for instance, to obtain the most stable structures of prototype nanotubes composed of particles interacting through a Lennard–Jones potential [26].

Boron two-dimensional structures [27] form foundations for BNTs since they can be obtained by “cutting out” adequate stripes and “gluing” them along the edges. Our previously developed methodology [28] based on the application of the floating-point representation is not the only possible strategy for the CSP of boron nanostructures. Alternatively, one can use binary (for monoatomic materials) or integer (for alloys) representations with a fixed crystal lattice. To obtain the total energy of the system, several approaches can be used. In our previous work, we used a genetic algorithm [29] combined with DFT to predict the structure of boron nanowires [28]. In this work, we will use a GA combined with a DFT-based nearest-neighbors model Hamiltonian to predict the structure of BNTs. We assume that the nanotubes have a perfect cylindrical shape and that their structure is closely related to one-atom-thick sheets of boron atoms arranged on a hexagonal lattice [27].

2. Computational Approach and Results of Simulations

2.1. Unit-Cell Definition

The information about the position of atoms and vacancies (“holes”) on a hexagonal lattice wrapped cylindrically around a nanotube axis is encoded using a binary representation. The type of considered nanotubes is limited to an armchair $(n, n)$ chirality as described in Ref. [8]. A cylindrical coordinate system can be conveniently used for the nanotube description. It is assumed that the nanotube has periodic boundary conditions (PBCs) in the z direction. The unit-cell size is described with two numbers $n_{\varphi },n_z\in {\mathbb{N}}_{+}$, where $n_{\varphi }>1$. The lattice constant (interatomic distance) of the unwrapped hexagonal layer is equal to a (to plot the BNTs in our figures, we have assumed $a=1.675$ Å). The positions of atoms and vacancies are not encoded in the genotype but are specified by the following position vectors:

$${\overrightarrow{r}}_i={\left(\frac{a\sqrt{6}}{4\sqrt{1-cos(\pi /n_{\varphi })}},\frac{\pi }{n_{\varphi }}·⌊\frac{i}{n_z}⌋,(imod{n}_z)·a+\left\{\begin{array}{cc}0\hfill & \mathrm{if}\lfloor i/n_z\rfloor \mathrm{is}\mathrm{even}\hfill \\ a/2\hfill & \mathrm{otherwise}\hfill \end{array}\right.\right)}_{(\rho ,\varphi ,z)}.$$

(1)

The use of binary representation, $g={\left(x_i\right)}_{i=0}^{c-1}\in {\mathbb{B}}^c$, where $c=2n_{\varphi }{n}_z$, justifies the way the vectors ${\overrightarrow{r}}_i$ are indexed (more details about indices is given in Appendix A). It also implies the number of atoms in the unit cell equal to $N_{\mathrm{a}}=\#\{i\in {\iota }_c\mid x_i\}$ and the number of vacancies equal to $N_{\mathrm{v}}=c-N_{\mathrm{a}}$.

The unit cell of the BNT is defined in terms of predicate theory. Predicate $Q\equiv Q_0\wedge Q_1\wedge Q_2$, where predicates $Q_0$, $Q_1$, and $Q_2$ state that atoms within the unit cell are connected, that there is an atomic neighborhood at the boundary of two unit cells along the nanotube axis, and that there is an atomic neighborhood at the unit cell boundary perpendicular to the circumference of the nanotube, respectively. The predicate Q is sufficient to ensure the nanotube connectivity and its closure at the circumference (see also Appendix A for a more precise formulation of the predicate Q). It does not restrict any possible valid result of the structures that are equivalent by translational or rotational symmetry.

2.2. Fitness Function

The fitness function values are calculated using a nearest-neighbors model Hamiltonian. The optimization task consists in finding such a configuration of atoms and vacancies, which maximizes the fitness function—the binding energy per atom, $E_{\mathrm{b}}$, calculated within the nearest-neighbors model Hamiltonian. In this model, $E_{\mathrm{b}}$ can be defined as:

$$E_{\mathrm{b}}(n_1,n_2,n_3,n_4,n_5,n_6)=\frac{1}{N_a}\sum _{i=1}^6{n}_i{e}_i,$$

(2)

where $n_i$ is the number of atoms in the unit cell with i nearest neighbors, while $e_i$ is the $E_{\mathrm{b}}$ for a structure consisting of atoms having only i nearest neighbors (e.g., the two-atom molecule, single atomic chain, or a honeycomb structure, hc, for i equal to 1, 2, or 3, respectively). Obviously, the total number of atoms in the unit cell $N_{\mathrm{a}}={\sum }_{i=1}^6{n}_i$. The values of $e_i$ are results of DFT calculations and are taken from our previous work [30]. They are equal to $e_1=1.7803$, $e_2=5.1787$, $e_3=5.6504$, $e_4=6.2522$, $e_5=6.5718$, and $e_6=6.5116$ in units of $\mathrm{eV}/\mathrm{atom}$. Since our model Hamiltonian includes only nearest-neighbors interactions and was developed for the case of a one-atom-thick layer of boron atoms [30,31], it does not take into account the curvature of the BNT. Therefore, the $e_i$ parameters can be applied to nanotubes with large radii, where the effect of curvature on the properties of the nanotube is expected to be negligible.

In our previous work [30], we have obtained for 2D boron structures that, on average, their $E_{\mathrm{b}}$ values calculated using Equation (2) differ from those obtained using DFT calculations by 45 and 54 meV/atom for buckled and flat structures, respectively. These errors are, however, about two times smaller than the energy difference, $\Delta E_{\mathrm{b}}^{\mathrm{DFT}}=91\mathrm{meV}/\mathrm{atom}$, between $E_{\mathrm{b}}$ values of 6.603 and 6.512 meV/atom for the $\alpha$-sheet and the buckled triangular ($bt$) sheet, respectively, and about twenty times smaller than the energy difference, $\Delta E_{\mathrm{b}}^{\mathrm{DFT}}=953\mathrm{meV}/\mathrm{atom}$, between $E_{\mathrm{b}}$ values of 6.603 and 5.650 meV/atom corresponding to the $\alpha$ and $hc$ sheets, respectively [30]. Similar accuracy may be expected for the $E_{\mathrm{b}}$ values predicted using Equation (2) for BNTs, at least for those nanotubes with large diameters (20 Å and above).

In the case of carbon, the elastic properties of the nanotubes do not depend on the diameter at approximately 18 Å [32]. The same is true for electronic properties such as the energy gap [33]. Making the assumption that the pure boron case situation is the same, $n_{\varphi }=23$ is taken, which corresponds to $\rho \approx 10.628$ Å. This gives the size of the potential solution space (described by a general formula of $2^{2n_{\varphi }{n}_z}$) equal to $2^{46n_z}$.

The Bernoulli $\mathrm{B}(1,0.5)$ probability distribution ($B(n,p)$ is a discrete probability distribution, where the Bernoulli random variable n can have only 0 or 1 as the outcome values, p is the probability of success ($n=1$), and $1-p$ is the probability of failure ($n=0$)) was chosen for the creation of the first generation of nanotubes, and genotypes not satisfying the predicate Q were rejected. A one-point recombination with bit-flipping mutation ($p=1/c$) applied stochastically with probabilities equal to $p_{\mathrm{r}}=p_{\mathrm{m}}=0.5$ is employed during the evolution.

The generation size $\mu$ is set to $4n_{\varphi }{n}_z$ while parent multiset size $2k=2n_{\varphi }{n}_z$. Stochastic universal sampling (SUS) with linear ranking selection ($s=2$) is used for the parent selection and the selection of the next-generation mechanisms. The genetic search is terminated after reaching a plateau for the fitness function or, more precisely, when after 100 generations the fitness function maximum has not improved by $\Delta E=1\mathrm{meV}$.

Some of the obtained nanotubes can be equivalent through rotations around the z-axis and translations along this axis. There is a relationship between natural numbers and the above-defined nanotubes. Encoding $g\in {\mathbb{B}}^{2n_{\varphi }{n}_z}$ corresponds to certain number $n={\sum }_{i=0}^{2n_{\varphi }{n}_z-1}{2}^i{x}_i$, where $\mathbb{B}\ni \mathrm{true}\equiv 1\in \mathbb{N}$ and $\mathbb{B}\ni \mathrm{false}\equiv 0\in \mathbb{N}$. There are classes of abstractions of natural numbers regarding rotations around the z-axis (nanotube axis) and translations along this axis, which form equivariant nanotubes. Rotations and translations in the set ${\iota }_{2n_{\varphi }{n}_z}$ can be defined as follows:

$$\begin{array}{cc}\hfill R_{n_{\varphi },n_z}^{\Delta n_{\varphi }}\left(n\right)& =n^{\prime },x_{(i+2\Delta n_{\varphi }{n}_z)mod2n_{\varphi }{n}_z}^{\prime }=x_i,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill T_{n_{\varphi },n_z}^{\Delta n_z}\left(n\right)& =n^{\prime },x_{\left((i+\Delta n_z)mod{n}_z\right)+\lfloor i/n_z\rfloor n_z}^{\prime }=x_i,\hfill \end{array}$$

(4)

where $\Delta n_{\varphi },\Delta n_z\in \mathbb{Z}$. The following identities hold:

$$\begin{array}{cc}\hfill R_{n_{\varphi },n_z}^{n_{\varphi }}& =\mathrm{id},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill T_{n_{\varphi },n_z}^{n_z}& =\mathrm{id},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill R_{n_{\varphi },n_z}^{\Delta n_{\varphi }}& ={\left(R_{n_{\varphi },n_z}^1\right)}^{\Delta n_{\varphi }},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill T_{n_{\varphi },n_z}^{\Delta n_z}& ={\left(T_{n_{\varphi },n_z}^1\right)}^{\Delta n_z},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill R_{n_{\varphi },n_z}^{-1}& =R_{n_{\varphi },n_z}^{n_{\varphi }-1},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill T_{n_{\varphi },n_z}^{-1}& =T_{n_{\varphi },n_z}^{n_z-1}.\hfill \end{array}$$

(10)

It may also turn out that the nanotube obtained as a result of evolution has a shorter elementary unit cell along ${\widehat{e}}_z$ than would result from the $n_z$ value. This occurs if and only if the maximum divisor $d_{n_z}$ of the number $n_z$, for which the condition

$$T_{n_{\varphi },n_z}^{n_z/d_{n_z}}\left(n\right)=n,$$

(11)

is satisfied for n representing a given nanotube, is greater than 1. If maximum divisor $d_{n_{\varphi }}$ of $n_{\varphi }$, for which the analogous condition

$$R_{n_{\varphi },n_z}^{n_{\varphi }/d_{n_{\varphi }}}\left(n\right)=n$$

(12)

is met, is greater than 1 then the nanotube has a nontrivial rotational symmetry about the axis of the tube. The values of $d_{n_z}$ and $d_{n_{\varphi }}$ are assumed to be the maximum values satisfying the conditions given by Equations (11) and (12), respectively.

2.3. Results of Simulations

The most stable evolutionary obtained nanotubes are labeled as ${\tau }_{n_{\varphi },n_z}$. The computational complexity for a given value of $n_{\varphi }$ grows exponentially with the unit cell length $n_z$. For $n_{\varphi }=23$, under the assumption that a single structure has proceeded for $10\mathsf{\mu }\mathrm{s}$ even for $n_z=2$ with sequential computations, the time five orders of magnitude longer than the Universe age [34] is needed to check all possible potential solutions. The application of a genetic algorithm is therefore advisable.

The most energetically favorable nanotubes are presented in Figure 1. The evolutionary obtained structure ${\tau }_{23,2}$ is the case of a fully filled with boron atoms lattice. The removal of any atom from a fully filled lattice is energetically favorable since it raises the number of 5-coordinated atoms (raise of the beneficial value of $n_5$). The density of vacancies cannot be too high since the situation with holes that are too close to each other may be detrimental for $E_{\mathrm{b}}$ as it may increase the number of 4-coordinated atoms (raise of the detrimental value of $n_4$). Looking at Figure 1, we may conclude that, in general, the most energetically favorable are those nanotubes for which the boron holes are separated by fragments of the boron double chain (BDC) or boron triple chain (BTC). Less favorable are those situations in which the holes are next to each other. The highest $E_{\mathrm{b}}$ value in our simulation belongs to the ${\tau }_{23,8}$ nanotube ($6.55108\mathrm{eV}/\mathrm{atom}$) after which the ${\tau }_{23,4}$ nanotube ($6.55102\mathrm{eV}/\mathrm{atom}$) is located. Recalling that DFT calculation precision is of the order of $1\mathrm{meV}/\mathrm{atom}$ [35], one can assume that both nanotubes have equal $E_{\mathrm{b}}$ values (within our model). In Figure 1, we also show the time taken to obtain the results, within our approach, for the most stable examples of BNTs for a given value of $n_z$. Although our methodology is relatively simple, the time taken to obtain, for instance, ${\tau }_{23,9}$ exceeds 6 days. This tells us that computations that would involve more accurate fitness functions may require prohibitive computational times.

The relation between $E_{\mathrm{b}}$ and hole density, $\eta =N_{\mathrm{v}}/c$, is presented in Figure 2. If one were to draw a curve that represents the average of the $E_{\mathrm{b}}$ values shown in Figure 2 as a function of hole density, the maximum of this curve would correspond to the hole density of $1/9=0.111$, which is the hole density of the $\alpha$-sheet [27] (the optimal flat neutral boron 2D structure). The histogram shown in the inset of Figure 2 presents the number of 4-, 5-, and 6-coordinated boron atoms as well as boron vacancies relative to the total number of possible boron sites ($c=N_{\mathrm{a}}+N_{\mathrm{v}}$) for the most stable BNTs. The average over the 7 relevant cases studied ($n_z\in \{3,4,\cdots ,9\}$) gives us that the number of 4-, 5-, and 6-coordinated boron atoms is about $2\%$, $63\%$, and $24\%$, respectively, of the total number of atoms forming the nanotubes, whereas about $11\%$ are boron vacancies. This gives us a very interesting result that more than $85\%$ of the boron atoms in BNTs with the largest $E_{\mathrm{b}}$ values are highly coordinated atoms.

In Figure 3, we show three examples of BNTs for three substantially different hole densities $\eta$ equal to 0, 0.122, and 0.239. For hole densities smaller than about 0.22, the boron nanotubes exhibit randomly distributed hexagonal holes and are more stable than a flat stripe structure and a quasi-flat B₃₆ cluster (see Figure 2). For larger hole densities (>0.22), the boron nanotubes resemble porous tubular structures with hole sizes that depend on the surface densities of boron atoms. Finally, it should be noted that almost all of the studied BNTs in this work have $E_{\mathrm{b}}$ values between $5.65\mathrm{eV}/\mathrm{atom}$ and $6.60\mathrm{eV}/\mathrm{atom}$, which correspond to $E_{\mathrm{b}}$ of the $hc$ and $\alpha$ sheets, respectively [30].

3. Summary

In conclusion, we have presented an adaptation of the genetic algorithm for the structure prediction of tubular BNTs in a broad range of possible hole densities. The evolution of the structures is achieved using a binary representation and a fitness function calculated using a model Hamiltonian. For the most stable BNTs, the number of 5-coordinated boron atoms is about $63\%$ of the total number of atoms forming the nanotubes, whereas about $11\%$ are boron vacancies. Moreover, more than $85\%$ of the boron atoms in BNTs with the largest $E_{\mathrm{b}}$ values are 5- and 6-coordinated atoms. For hole densities smaller than about 0.22, the BNTs exhibit randomly distributed hexagonal holes and are more stable than the BTC structure and a quasi-flat B₃₆ cluster. For larger hole densities (>0.22), the BNTs resemble porous tubular structures with hole sizes that depend on the surface densities of boron atoms. These prototype genetic algorithm calculations may serve for the exploration of the structure of nanotubes based on a mixture of boron and other atoms (e.g., carbon and/or nitrogen).

In addition, we can learn from our approach that the central parameters that determine the structure of BNTs are the coordination numbers of the constituent atoms. Each atom that has a given coordination number contributes with a given energy to $E_{\mathrm{b}}$, and from the balance between different contributions we obtain porous BNTs with regularly distributed holes. We do not observe the fragmentation of the nanotubes for any of the studied hole densities. This remarkable result should attract the attention of experimentalists seeking to extend the effort of synthesizing BNTs to structures with a broader range of hole densities (from 0 to up to about 0.3), which would certainly extend a range of possible applications of BNTs.