# The structure of thin boron nanowires predicted using evolutionary
computations

Tomasz Tarkowski  Nevill Gonzalez Szwacki  Institute of Theoretical Physics,
Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warszawa,
Poland  Institute of Experimental Physics, Faculty of Physics, University of
Warsaw, Pasteura 5, PL-02093 Warszawa, Poland

###### Abstract

This work describes the implementation of a genetic algorithm-based strategy
combined with first-principles computations for identifying the structure of
the most stable boron 1D structures. We focus our attention on the structure
of ultrathin 1D boron structures given the lack of previous experimental and
theoretical work on this topic. Our methodology yields reasonable structural
candidates for further optimizations at the DFT level with tighter convergence
criteria. The simulations involved 1D structures with up to 8 atoms per unit
cell. We have identified four main groups of structures: flat nanowires
(monatomic-height stripes) with triangular or triangular and “square” motifs,
stripes with larger holes, nanowires with an open tubular shape, and regular
nanowires. The diameter-dependent structural changes are discussed.

###### keywords:

genetic algorithms, first-principles calculations, boron nanowires, boron 1D
structures

††journal: Solid State Sciences

##  1 Introduction

Genetic algorithms (GAs) serve as a framework for solving optimization
problems. One such problem is crystal structure prediction (CSP) based solely
on stoichiometry. Brute force CSP is prohibitively expensive [1] and often
lacks proper solution [1, 2].

Boron is one of the least understood chemical elements in the periodic table
[3]. It was discovered in 1808 [4] and first obtained in pure form in 1909
[5]. Ultrapure boron single crystals for the semiconductor industry are
obtained from the decomposition of diborane assisted with Czochralski or zone
melting purification processes [6]. Boron has two naturally occurring
isotopes, ${}^{10}{\rm B}$ and quantitatively dominant ${}^{11}{\rm B}$ (both
stable) [7], and over a dozen radioisotopes up to ${}^{21}{\rm B}$ [8], which
in this particular example decays through a two-neutron emission channel.
Boron in the Universe and on the Earth comes from the spallation process [9].
Its abundance in the continental crust is at the level of $0{.}001\%$, which
ranks this element at position number 37, after lead and before thorium [10].
Boron is important to life on Earth—it co-creates plant cell walls [11]. This
element was discovered on Mars in Gale Crater in way suggesting that water was
there in the past [12]. On Earth, on the west edge of the Mojave Desert (USA),
in a census-designated place with not accidental name of Boron (California,
USA) is located the greatest borax (${\rm Na}_{2}{\rm B}_{4}{\rm
O}_{7}\\!\cdot\\!10{\rm H}_{2}{\rm O}$) mine on the planet. Borax can be
easily converted into boric acid and then into boron trioxide, and finally
into elemental boron.

Boron has an electron configuration of $[{\rm He}]\ 2s^{2}\ 2p^{1}$, which
means that it cannot form carbon-like hybridization. Instead, it can form e.g.
covalent bonds with three centers sharing two electrons (3c-2e) [13]. All
known three dimensional (3D) boron allotropes are based on 12-atom icosahedra
${\rm B}_{12}$ [14, 15, 16].

Similarly to carbon, boron can form low-dimensional structures. In the two-
dimensional (2D) case, theoretically, the most stable neutral structure is the
$\alpha$-sheet [17], however in electron-rich environments, e.g., during
production on metallic substrates, the negatively charged $\delta$-sheet would
be preferred [18]. This structure was obtained experimentally on a silver
substrate by molecular beam epitaxy (MBE) [19]. The first 2D boron structures
were obtained in 2015 chemical vapor deposition (CVD) [20] and MBE [21]. 2D
boron production is now standard procedure [22, 23, 24].

In the one-dimensional (1D) case, boron forms nanowires [25, 26, 27, 28, 29]
and nanotubes [29, 30, 31, 32, 33]. 1D boron of the BDC (_boron double chain_)
type (cf. $s_{4}$ structure from Ref. [34]) is metallic and so robust that it
does not undergo Peierls transition [35] whereas it has a Kohn anomaly in the
phonon dispersion relation [36]. 1D boron of the BTC (_boron triple chain_)
type is energetically more favorable than the BDC type [34].

Zero-dimensional (0D) boron forms structures similar to carbon fullerenes,
e.g., ${\rm B}_{80}$ (_buckyball_) [37] and ${\rm B}_{40}$ (_borospherene_)
molecules [38]. Experimental and theoretical research on ${\rm B}_{36}$
cluster [39] formed the basis for obtaining 2D boron structures.

Boron can form materials with other chemical elements (alloys). One such
example is boron nitride nanotube (BNNT) [40, 41]. Paper made of BNNTs,
contrary to ordinary paper or made of carbon nanotunes, is flame resistant
[42]. BNNTs can also spontaneously reconstruct after bending [43].

Scientific and engineering breakthroughs of the next two decades, especially
the increase in computing power, enable—at least partially—the realization of
CSP. At the beginning of this century, Oganov et al. [1] realized a computer
program called USPEX, which merges the GA approach with density functional
theory (DFT) calculations. This method, despite being computationally quite
expensive, is fruitful—boron allotrope $\gamma$-${\rm B}_{28}$ [44, 45] and
high-pressure sodium [46] are two examples where numerical simulations were
experimentally confirmed.

The USPEX program is not the first case of an evolutionary computation (EC)
software package applied to the structure prediction task, a similar approach
was previously used for the protein folding problem [47]. On the other hand,
it seems that Oganov’s work was the first attempt which encouraged the
condensed matter physics community for this method—sometime later Zurek et al.
[48] presented the XtalOpt program that is using similar GA/DFT combination,
while scientists working on the CALYPSO project [49, 50] implemented the
particle swarm optimization (PSO) algorithm to determine the crystal structure
of materials.

Boron in its 1D form remains the least studied, both theoretically and
experimentally [3, 51]. In this work, we find the structure of boron nanowires
with diameters smaller than 1 nm. We are interested to study the structural
changes of these ultrathin nanowires as a function of their diameter. To
achieve this goal, the most stable forms of all-boron nanowires are found
using a developed for this purpose software package called Quilë [52].

##  2 Theoretical approach

An evolutionary algorithm is a non-deterministic optimization method that in
each run can generate different solutions, even if the initial structures are
the same. The task to be solved in this work is to find—with the use of GA and
DFT—the optimal crystal structure of monoatomic nanowires. Two groups of 1D
structures will be considered. In the first group, the boron structures are
confined to planarity. This kind of structures are commonly known as atomic
stripes. In the second group of structures, the out of plane position of atoms
is allowed and as a result of the structure optimization we can end up having
buckled stripes, nanowires, and even nanotubes. The first step in our nanowire
CSP’s problem is to define the unit cell in terms of predicate formalism.

###  2.1 Unit cell definition in terms of predicate $Q$

The unit cell of our 1D structure is defined in terms of predicate $Q$.
Without loss of generality one can make the following assumptions:

  * 1.

the 1D structures are periodic in the $z$ direction,

  * 2.

the length of the unit cell in the $z$ direction is equal to $h$,

  * 3.

atomic positions in the unit cell are described with a sequence of vectors
$(\vec{r}_{i})_{i=0}^{N_{\rm a}-1}$ fulfilling the condition $\forall
i,j\in\iota_{N_{\rm a}}\colon(i<j\Rightarrow z_{i}\leq
z_{j})\wedge(i=j\Leftrightarrow\vec{r}_{i}=\vec{r}_{j})$, where $N_{\rm a}$ is
the number of atoms in the unit cell,

  * 4.

the first atom in the unit cell is at the origin of the coordinate system,
i.e., $\vec{r}_{0}=\vec{0}$,

  * 5.

flat structures are described using Cartesian coordinates $(0,y,z)$, while
nanowires are described using cylindrical coordinates $(\rho,\phi,z)$, where
we choose $\phi_{1}=0$.

It can be easily seen that for flat structures $N_{\rm a}\geq 1$, while for
nanowires $N_{\rm a}\geq 3$.

The $Q$ predicate should state two things ($Q\Leftrightarrow Q_{0}\wedge
Q_{1}$). Firstly, no two atoms should be too close to each other ($Q_{0}$
predicate). Secondly, for each atomic pair, the path of interatomic bonds
connecting this pair should exist ($Q_{1}$ predicate). In order to fulfill
periodic boundary conditions both predicates (i.e. $Q_{0}$ and $Q_{1}$) should
be formulated based on two adjacent unit cells. These conditions demand the
use of two parameters, labeled as $d_{\min}$ and $d_{\max}$, describing the
allowed range of interatomic distances, beyond which the bond cannot be formed
because of too small or too big interatomic distance, respectively. The
$d_{\min}$ and $d_{\max}$ parameters should be chosen for each material
separately and _a priori_ knowledge of the chemistry of the material might be
helpful to constrain the permitted range of distances.

The $Q_{0}$ predicate can be formulated as follows:

$\displaystyle Q_{0}\equiv\forall i,j\in\iota_{N_{\rm a}}\colon$
$\displaystyle\left(i\neq j\Rightarrow\left|\vec{r}_{i}-\vec{r}_{j}\right|\geq
d_{\min}\right)\wedge\left(\left|\vec{r}_{i}-\left(\vec{r}_{j}+h\hat{e}_{z}\right)\right|\geq
d_{\min}\right)$ $\displaystyle\Leftrightarrow\forall i,j\in\iota_{N_{\rm
a}}\colon$ $\displaystyle\left[i\neq
j\Rightarrow\left(\left|\vec{r}_{i}-\vec{r}_{j}\right|\geq
d_{\min}\wedge\left|\vec{r}_{i}-\left(\vec{r}_{j}+h\hat{e}_{z}\right)\right|\geq
d_{\min}\right)\right]\wedge h\geq d_{\min}.$ (1)

To formulate the $Q_{1}$ predicate, elements of graph theory will be used. Let
$V$ be the set of _vertices_ consisting of atomic positions taken from two
adjacent unit cells, i.e., $V=\\{\vec{r}_{i},\vec{r}_{i}+h\hat{e}_{z}\mid
i\in\iota_{N_{\rm a}}\\}$. Let $E$ be the set of _edges_ (undirected), which
vertices fulfill the condition of maximum distance $d_{\max}$:
$E=\\{\\{\vec{q},\vec{r}\\}\subseteq
V\mid\vec{q}\neq\vec{r}\wedge|\vec{q}-\vec{r}|\leq d_{\max}\\}$. By taking a
pair of both objects one obtains _graph_ $H=(V,E)$. The $Q_{1}$ predicate
might be now formulated as $Q_{1}\equiv\omega(H)=1$, where $\omega(H)$ is the
number of _connected components_ of graph $H$, i.e., as a statement that graph
$H$ is connected. Graph connectivity can be checked with depth- or breadth-
first search, or by analyzing the adjacency matrix. It should be noted that
the aforementioned $Q_{1}$ predicate should be treated as a first
approximation of the solution, i.e., every nanowire representation fulfilling
$Q_{1}$ predicate is proper (nanowire is connected), but there are some proper
nanowire representations, for which $Q_{1}$ predicate does not hold (“false
negative”). This approximation of $Q_{1}$ is computationally cheaper than its
exact version with checks employing translations along the $z$ axis.

###  2.2 Encoding of the unit cell traits. Fitness function

The information about the phenotype, i.e., atomic positions and unit cell
height, should be somehow encoded in the genotype. For this purpose a
floating-point representation in the form of
$g=(x_{i})_{i=0}^{c-1}\in[0,1]^{c}\subset\mathbb{R}^{c}$ is used. The
genotypes for flat and non-flat cases are shown in Fig. 1, where the scheme of
encoding between the phenotype and genotype is sketched.

Figure 1: Genotypes for flat and non-flat nanowires with encoded phenotypic
traits. The $x_{j}$ gene can encode the $\vec{r}_{i}$ coordinate of atom
number $i$ or unit cell height $h$. The unit cell height is encoded by
multiple genes (polygene). Single gene $x_{2}$ defines multiple phenotypic
traits, i.e., $z_{i}$ coordinates of all atoms for $i>0$ and height $h$
(pleiotropy).

One can notice that the genotype length, $c$, depends on the nanowire type:

$c=\left\\{\begin{array}[]{ll}2N_{\rm a}-1&{\rm flat},N_{\rm a}\geq 1\\\
3(N_{\rm a}-1)&{\rm non-flat},N_{\rm a}\geq 3\\\ \end{array}\right..$ (2)

The same is for the unit cell height, $h$, and atomic positions
$(\vec{r}_{i})_{i=0}^{N_{\rm a}-1}$. For flat nanowires
$h=d_{\max}(x_{0}+\sum_{j=1}^{N_{\rm a}-1}x_{2j})$ and the atomic positions
for $i\geq 1$ in Cartesian coordinates are:

$\vec{r}_{i}=d_{\max}\left(0,(N_{\rm
a}-1)\left(\frac{1}{2}x_{2i-1}-1\right),\sum_{j=1}^{i}x_{2j}\right).$ (3)

For non-flat nanowires $h=d_{\max}(x_{0}+\sum_{j=1}^{N_{\rm a}-1}x_{3j-1})$
and the atomic positions for $i\geq 1$ in cylindrical coordinates are:

$\vec{r}_{i}=d_{\max}\left((N_{\rm
a}-1)\cdot\left\\{\begin{array}[]{l}x_{1}\\\ x_{3i-3}\\\
\end{array}\right.,\,\frac{2\pi}{d_{\max}}\cdot\left\\{\begin{array}[]{l}0\\\
x_{3i-2}\\\
\end{array}\right.,\,\sum_{j=1}^{i}x_{3j-1}\right)_{(\rho,\phi,z)}\left\\{\begin{array}[]{l}i=1\\\
i\geq 2\\\ \end{array}\right..$ (4)

The optimization problem is to find such unit cell height and atomic positions
which minimize the total energy. This energy is obtained from self-consistent
DFT calculations using the Quantum ESPRESSO (QE) software package [53, 54].

###  2.3 The case of boron nanowires

The self-consistent DFT calculations are performed using the projector
augmented wave (PAW) technique and the Perdew–Burke–Ernzerhof (PBE) exchange-
correlation functional. A Monkhorst–Pack Brillouin zone sampling with a
$1\times 1\times 8$ $k$-point mesh is used with a half-step offset in the $z$
direction. Broyden electron density mixing with mixing parameter $\beta=0{.}7$
and a Methfessel-Paxton method with a smearing width of $0{.}01\,{\rm Ry}$ for
the electron smearing are used. The kinetic energy cut-off of the plane wave
basis is set to $60\,{\rm Ry}$. The PAW pseudopotential is taken from the QE
library [55]. The projection of the distance between extreme atoms of two
adjacent unit cells on $x$ and $y$ axes (_buffer thickness_) is equal to
$10\,\text{\AA}$, which should be enough to minimize the effects of nanowire
replicas placed periodically in the $xy$-plane.

For the purpose of random crystal generation [56] at the first GA iteration
(first generation of genotypes), a mechanism of drawing genes’ values from a
uniform distribution is used. The genotypes violating the $Q$ predicate are
rejected and the drawing procedure is repeated until $Q$ is fulfilled. Single
arithmetic recombination with a random-reset mutation with parameter $p=1/c$
applied stochastically is used with recombination and mutation probabilities
equal to $p_{\rm r}=1$ and $p_{\rm m}=0{.}5$, respectively. The generation
size $\mu$ is set to $100$, while the parents’ multiset size is set to
$2k=64$. The stochastic universal sampling (SUS) mechanism with linear ranking
selection (lin-RS, $s=2$) is used for parents’ selection and selection to the
next generation. The genetic search is finished after reaching the fitness
function _plateau_ , or—more precisely—if after $10$ last generations the
fitness function maximum has not improved more than $\Delta E$ with respect to
the whole process. The $\Delta E$ value depends on the number of atoms in the
unit cell: $\Delta E=\Delta E_{\rm b}\cdot N_{\rm a}$, where $\Delta E_{\rm
b}=1\,{\rm meV}/{\rm atom}$ is a DFT precision of binding energy per atom
calculations resulting from the assumed approximations [57]. The bond lengths
are constrained between $d_{\min}=1{.}54\,\text{\AA}$ and
$d_{\max}=2{.}10\,\text{\AA}$ inclusively—these values arise from statistical
analysis of known boron structures [58].

A complete implementation of boron nanowires CSP is available as an example
program in the directory _examples/evenstar/_ of the Quilë repository.
Numerical simulations for flat and non-flat nanowires for $N_{\rm a}$ up to
$8$ are performed with the above-mentioned recipe. In order to avoid potential
non-full effectiveness of the genetic search, calculations for each $N_{\rm
a}$ case, for both flat and non-flat nanowires, are performed $5$ times and
calculation logs are available online [59] (please see _evolution/_
directory).

The structures obtained from our GA search are further optimized using QE.
Only one or, in case of degeneracy into the same energy, several best
structures are further proceeded. For each of these structures, we have
optimized the atomic positions and the lattice constants with an increase to a
$1\times 1\times 16$ $k$-point grid and an increase to a $15\,\text{\AA}$
buffer thickness. The total energy ($E_{\rm tot,\ isolated\ B\ atom}$) for an
isolated boron atom is obtained with calculations including spin
polarization—boron has an unpaired electron on valence shell (results of
calculations are available online [59] in the _single_atom/_ directory). The
results of simulations are visualized with the use of the XCrySDen program
[60]. Animated visualization of nanowires is presented online [59] in the
_results/_ directory, where all structures are presented, both before and
after relaxation.

Figure 2: The structure of flat and non-flat nanowires obtained by
evolutionary computations followed by DFT relaxation. In both cases, the
following data is shown (from top): number of atoms in the unit cell $N_{\rm
a}$, evolutionary found structure, the consequently DFT-relaxed structure, the
structure label (symbols $\delta_{N_{\rm a}}^{i}$ or $\kappa_{N_{\rm a}}^{i}$
for $i\in\mathbb{N}$), and the binding energy per atom, $E_{\rm b}$ (in units
of ${\rm eV}/{\rm atom}$), for the DFT-relaxed nanowires. Structures with
labels in bold have the highest $E_{\rm b}$ from all nanowires with a given
number of atoms in the unit cell. For the non-flat nanowires, beside the side
view (alongside the $z$ axis), the top view is shown preserving the scale.
Note that the scale between different structures is not preserved.

##  3 Results and discussion

The collection of the evolutionary found and DFT-relaxed 1D structures is
shown in Fig. 2. They are divided into two groups: nanowires obtained through
GA-evolution of flat $\delta_{N_{\textnormal{a}}}^{i}$ (gr. διαµεριςµα – flat)
and non-flat $\kappa_{N_{\textnormal{a}}}^{i}$ (gr. ϰυρτος – convex)
structures, where $i\in\mathbb{N}$ is an index labeling structures
($\delta_{N_{\textnormal{a}}}^{*}$ or $\kappa_{N_{\textnormal{a}}}^{*}$)
having the same number of atoms in the unit cell. The DFT-relaxed structures
are presented in Fig. 2 below the corresponding evolutionary found
counterparts. The most energetically favorable nanowires within each group
($\delta_{N_{\textnormal{a}}}^{*}$ and $\kappa_{N_{\textnormal{a}}}^{*}$) are
highlighted in the figure using bold labels. The binding energy, $E_{\rm b}$,
is calculated using the formula:

$E_{\rm b}=E_{\rm tot,\ isolated\ B\ atom}-\frac{E_{\rm tot,\
nanowire}}{N_{\rm a}}.$ (5)

Analyzing the results presented in Fig. 2, we can identify two groups of flat
structures: stripes with triangular or triangular and “square” motifs, and
stripes with larger holes. Looking at non-flat structures, we can find, among
others, nanowires with an open tubular shape (e.g. $\kappa_{5}^{0}$,
$\kappa_{6}^{1}$, or $\kappa_{8}^{1}$) and regular nanowires (e.g.
$\kappa_{4}^{0}$, $\kappa_{5}^{2}$, or $\kappa_{6}^{2}$). The $\delta_{2}^{0}$
and $\kappa_{4}^{1}$ stripes represent the same structure. Doubling the unit
cell size (in the $z$ direction) does not result in any structural changes
what confirms the robustness of the BDC structure against Peierls distortions.
The $\delta_{3}^{0}$ and $\delta_{4}^{0}$ stripes are also found in the
$\kappa_{3}^{*}$ (and also $\delta_{6}^{*}$) and $\kappa_{4}^{*}$ (and also
$\delta_{8}^{*}$) series of structures, respectively, and are shown only once
in Fig. 2. The widest considered 1D structure is $\delta_{7}^{3}$ (9 Å in
diameter) and the thinnest is the boron chain $\delta_{1}^{0}$.

Figure 3: Binding energy per atom, $E_{\rm b}$, for evolutionary obtained and
then DFT-relaxed flat (left panel) and non-flat (right panel) nanowires from
Fig. 2 as a function of the number of atoms in the unit cell, $N_{\rm a}$. The
values of $E_{\rm b}$ for two-dimensional triangular structures, flat
$s_{6}$-ft and buckled $s_{6}$-bt, marked with dashed lines are taken from
[61]. Each panel includes three insets presenting: an average number of
generations across 5 computational series for each $N_{\rm a}$ value (top
inset), the respective average computational time of the GA-evolution on a
single CPU [62] machine (bottom left inset), and normalized energy difference
$\Delta E_{\rm b}/E_{\rm b,\ final}$, where $\Delta E_{\rm b}=E_{\rm b,\
final}-E_{\rm b}$, as a function of the generation number (bottom right inset)
during GA-evolution for the most energetically favorable structures,
$\delta_{3}^{0}$ and $\kappa_{8}^{1}$.

From small-diameter structures, the most stable are fully planar stripes with
boron triangular motifs. Among all flat structures, the largest value of
$E_{\rm b}$ belongs to the $\delta_{3}^{0}$ nanowire ($5{.}817\,{\rm eV}/{\rm
atom}$) while among non-flat structures—to the $\kappa_{8}^{1}$ nanowire
($5{.}840\,{\rm eV}/{\rm atom}$). Boron can also form nanotubes and the open
shape of $\kappa_{8}^{1}$ suggests it as well. The value of $E_{\rm b}$ for
$\kappa_{8}^{1}$ is close to that of the buckled triangular boron sheet (see
Fig. 3).

The values of $E_{\rm b}$ for 1D structures are usually lower than those for
2D structures of the same material. It is worth noting that $E_{\rm b}$ for
some of the obtained boron nanowires is larger than that for 2D boron
triangular flat sheet ($s_{6}$-ft) [61]. This is shown in Fig. 3 where we plot
$E_{\rm b}$ for each structure from Fig. 2 as a function of $N_{\rm a}$.
Figure 3 also shows three insets presenting the average number of generations
across 5 computational series for each $N_{\rm a}$ value, the respective
average computational time of GA-evolution on a single CPU [62] machine, and
normalized energy difference $\Delta E_{\rm b}/E_{\rm b,\ final}$ as a
function of the generation number during GA-evolution resulting in the most
energetically favorable structures, $\delta_{3}^{0}$ and $\kappa_{8}^{1}$. For
both flat and non-flat structures, the number of generations needed to obtain
the results increases with the number of atoms in the unit cell. On the other
hand, at the logarithmic scale, the average computational time increases
linearly with $N_{\rm a}$ for flat structures and rises rapidly for non-flat
structures.

##  4 Summary

In summary, we study the structure of ultrathin boron 1D periodic structures
using a DFT-based GA approach. Based on our simulations, we have identified
four main groups of structures: stripes with triangular or triangular and
“square” motifs, stripes with larger holes, nanowires with an open tubular
shape, and regular nanowires. The most stable thin structures are fully planar
stripes with boron triangular motifs. However, thicker stripes tend to form
tubular-like structures that can be viewed as precursors of nanotubes or
nanowires. Finally, several of the studied 1D structures are more stable than
the boron hexagonal (triangular) sheet which gives hope that such thin
structures may be experimentally obtained. Although the whole methodology
developed in this work is tested on boron 1D structures, we believe that it
has general predictive capabilities and can be used to study e.g. nanowires
made of silicon or other monoatomic or polyatomic materials.

## Acknowledgments

This work is a result of the projects funded by the National Science Centre of
Poland (Twardowskiego 16, PL-30312 Kraków, Poland, http://www.ncn.gov.pl/)
under the grants number UMO-2013/11/B/ST3/04273 and UMO-2016/23/B/ST3/03575.

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