Abstract
It remains rather difficult for traditional computational methods to reliably predict the properties of nanosystems, especially for those possessing pronounced radical character. Accordingly, in this work, we adopt the recently formulated thermallyassistedoccupation density functional theory (TAODFT) to study twoatomwide linear boron nanoribbons lBNR[2,n] and twoatomwide cyclic boron nanoribbons cBNR[2,n], which exhibit polyradical character when the n value (i.e., the number of boron atoms along the length of lBNR[2,n] or the circumference of cBNR[2,n]) is considerably large. We calculate various electronic properties associated with lBNR[2,n] and cBNR[2,n], with n ranging from 6 to 100. Our results show that lBNR[2,n] and cBNR[2,n] have singlet ground states for all the n values examined. The electronic properties of cBNR[2,n] exhibit more pronounced oscillatory patterns than those of lBNR[2,n] when n is small, and converge to the respective properties of lBNR[2,n] when n is sufficiently large. The larger the n values, the stronger the static correlation effects that originate from the polyradical nature of these ribbons. Besides, the active orbitals are found to be delocalized along the length of lBNR[2,n] or the circumference of cBNR[2,n]. The analysis of the sizedependent electronic properties indicates that lBNR[2,n] and cBNR[2,n] can be promising for nanoelectronic devices.
Introduction
Boron is as versatile as carbon in forming different nanostructures. Over the past few decades, boron nanomaterials have been explored extensively, and there has been growing interest in the investigation of boron nanomaterials, due to their interesting properties and potential applications in electronics and other industries^{1,2,3,4,5,6,7,8}. The recent interest in boron and other nanomaterials has been partly inspired by the structures, properties, and applications of carbon nanomaterials (e.g., the C_{60} fullerene, carbon nanotubes (CNTs), and graphene)^{9,10,11}. The prediction of B_{80} fullerene^{12}, observation of B_{40} fullerene^{13}, and observation of the Dirac cone in borophene^{9,14,15} have also shown promise in electronics and industrial applications. Besides, boron nanomaterials have also been proposed for supercapacitor^{16}, Liion battery^{17,18}, molecular Wankel motor^{6}, and hydrogen storage^{19,20} applications.
The vast boron nanomaterials can be planar or quasiplanar with single or multiple hexagonal vacancies, tubular, etc. The clusters can be neutral, cationic, or anionic. Among various boron nanomaterials, onedimensional (1D) and quasi1D nanostructures^{4,5,8,21,22,23,24} are of great interest for nanoelectronics applications. In the case of 1D boron nanostructures, the unique properties of CNTs have motivated interest in boron nanotubes^{4}. For CNTs, the chiralitydependent electronic properties are interesting, but it remains challenging to synthesize CNTs with uniform chirality, which is necessary for electronics applications. Thus, novel 1D structures based on boron have also been intensively studied in recent years. With the theoretical predictions of boron fullerenes and 2D sheets, obtaining the possible structures and properties of 1D and quasi1D boron nanomaterials can be the obvious next step.
Very recently, Liu et al.^{24} studied the mechanochemical properties of 1D boron chains (also known as linear boron chains) and quasi1D boron nanoribbons (also known as twoatomwide linear boron nanoribbons (which can be the narrowest linear boron nanoribbons)). The study revealed that the quasi1D boron nanoribbons are more stable than the 1D boron chains. These boron chains and nanoribbons exhibit attractive mechanochemical property (i.e., stressdependent structural transition between the quasi1D boron nanoribbons and 1D boron chains). The quasi1D boron nanoribbons are metallic in equilibrium. However, when they are stretched, the quasi1D boron nanoribbons can morph into the antiferromagnetic semiconducting 1D boron chains. On the other hand, when the stretched 1D boron chains are released, they can fold back into the metallic quasi1D boron nanoribbons. The 1D boron chains and quasi1D boron nanoribbons have very high mechanical stiffness of 46 to 72 eV/Å. These interesting mechanochemical properties make these boron chains and boron nanoribbons potential materials for constantforce springs at the nanoscale^{24}.
As illustrated in Fig. 1(a), a twoatomwide linear boron nanoribbon with n boron atoms along the length of ribbon, which is designated as lBNR[2,n], is studied in the present work. Besides, its cyclic isomer, a twoatomwide cyclic boron nanoribbon with n boron atoms along the circumference of ribbon (see Fig. 1(b)), which is denoted as cBNR[2,n], is also studied here. Note that cBNR[2,n] may find applications in molecular motors, nanoscale devices, electronics, etc. In addition, cBNR[2,n] can be regarded as the building blocks of boron nanotubes. It has been reported that boron αsheets are metallic. However, when they are wrapped to form singlewalled boron nanotubes, the latter (with a diameter less than 20 Å) become semiconducting. The metaltosemiconductor transition is due to the curvatureinduced surface buckling^{25}. The experimentally observed boron nanowires with diameters ranging from 20 to 200 nm and lengths up to a few μm have been found to be semiconducting in nature^{26}. Therefore, a comprehensive study on the electronic properties of lBNR[2,n] and cBNR[2,n] may provide further insight into the development and applications of boron nanotubes as well.
As of now, the studies of lBNR[2,n]^{24,27,28} and cBNR[2,n]^{29,30,31} remain very scarce. Recent computational studies have shown that B_{n}\({{\rm{H}}}_{2}^{2}\) (up to n = 22)^{32}, Li_{2}B_{n}H_{2} (up to n = 22)^{32}, and B_{12}\({{\rm{F}}}_{n}^{0/}\) (n = 1–6)^{33} have structures similar to lBNR[2,n]. Experimentally, a photoemission spectroscopy study has revealed the presence of dihydrogenated boron clusters H_{2}\({{\rm{B}}}_{n}^{}\) (n = 7–12), which also have structures similar to lBNR[2,n]^{27}. Besides, crystalline boron nanoribbons with widths ranging from 20 to 100 nm have been successfully synthesized and characterized by Xu et al.^{3}. However, it remains very difficult to synthesize lBNR[2,n]. On the other hand, while singlewalled boron nanotubes have been successfully synthesized^{34,35}, the syntheses of cBNR[2,n] have not been realized yet. The difficulty in synthesizing lBNR[2,n] and cBNR[2,n] may be due to the presence of strong static correlation effects in these materials (commonly occurring in lowdimensional structures because of the effect of quantum confinement^{36}). Therefore, a computational study on the electronic properties of lBNR[2,n] and cBNR[2,n] with various n values, can be essential for the progress in this field, and may also play an important role in the selection of ideal materials for nanoelectronics applications.
Among various computational methods, presently, KohnSham density functional theory (KSDFT)^{37} remains very popular, because of the desired balance between accuracy and efficiency. Nonetheless, KSDFT with approximate exchangecorrelation (XC) density functionals can have spectacular failures (e.g., the selfinteraction error, static correlation error, etc.) in certain situations^{38,39}. In particular, KSDFT with conventional semilocal, hybrid, and doublehybrid XC density functionals cannot adequately describe the groundstate properties of systems with radical nature, such as the larger lBNR[2,n]/cBNR[2,n] (as will be shown later). Typically, accurate multireference (MR) computational approaches are required for the study of systems with radical nature^{40,41,42,43,44,45}. Nevertheless, due to the expensive computational cost, calculations based on accurate MR computational approaches are applicable only for small systems, and become intractable for large systems (especially for geometry relaxation). Accordingly, nanosystems with radical nature are beyond the reach of traditional computational methods. Since the number of electrons in lBNR[2,n]/cBNR[2,n] quickly increases with the system size (n), calculations based on the presently available MR computational approaches are unlikely to be feasible, especially for the larger lBNR[2,n]/cBNR[2,n].
Aiming to resolve the aforementioned problem, we have developed TAODFT (thermallyassistedoccupation density functional theory)^{46} for the study of nanosystems with radical nature in recent years. Note that TAODFT, which employs fractional orbital occupations generated by the FermiDirac distribution (governed by a fictitious temperature θ), reduces to KSDFT in the absence of strong static correlation, allowing a wellbalanced description for both systems with nonradical nature and systems with radical nature. Within the framework of TAODFT, the presently available local density approximation (LDA)^{46}, generalizedgradient approximation (GGA)^{47}, and global hybrid^{48} XC density functionals can also be adopted. Besides, we have also developed a scheme for the selfconsistent determination of θ to improve the overall accuracy of TAODFT for general applications^{49}. To demonstrate its applicability, we have recently employed TAODFT for the study of the electronic properties of several nanosystems with radical nature, including acenes^{46,47,48}, zigzag graphene nanoribbons^{50}, cyclacenes^{51}, Möbius cyclacenes^{52}, alternant polycyclic aromatic hydrocarbons^{53}, and the coronene series^{54}. Besides, we have also employed TAODFT to search for desirable hydrogen storage materials among nanosystems with radical nature in recent years^{19,55,56}. Very recently, TAODFT and related methods have also been successfully applied to study the electronic properties of cyclic nanorings and singlewalled CNTs by other research groups^{57,58}. Accordingly, in this work, we employ TAODFT to study the electronic properties of lBNR[2,n] and cBNR[2,n], with n ranging from 6 to 100.
Computational Details
We perform all calculations with QChem 4.4^{59}, using the 6–31 G(d) basis set and the numerical grid containing 75 EulerMaclaurin radial grid points and 302 Lebedev angular grid points. Results are obtained from TAOLDA^{46} (i.e., TAODFT with the LDA exchange, correlation, and θdependent density functionals) with the fictitious temperature θ = 7 mhartree.
While more complicated XC functionals (e.g., the GGA^{47} and global hybrid^{48} XC functionals) may be employed in TAODFT as well, they outperform TAOLDA primarily for the properties closely related to shortrange XC effects (e.g., the atomization energies and barrier heights of systems with nonradical nature), not for the properties closely related to static correlation (e.g., the singlettriplet energy gaps and fundamental gaps of systems with radical nature)^{46,47,48}. For example, the GGA and global hybrid XC functionals in TAODFT were found to perform similarly to TAOLDA for the electronic properties of linear acenes (i.e., systems with polyradical nature)^{46,47,48}. Consequently, the electronic properties of lBNR[2,n] and cBNR[2,n] from TAOLDA should be qualitatively similar to those from the GGA and global hybrid XC functionals in TAODFT.
It is worth mentioning that for TAOLDA, the vertical ionization potential, vertical electron affinity, and fundamental gap of a molecule cannot be directly calculated using the negative of the highest occupied molecular orbital (HOMO) energy, the negative of the lowest unoccupied molecular orbital (LUMO) energy, and the HOMOLUMO gap (i.e., the energy difference between the HOMO and LUMO), respectively, due to the possibility of fractional orbital occupations (see, e.g., Section III of ref.^{49}). Therefore, in this work, the vertical ionization potential, vertical electron affinity, and fundamental gap of lBNR[2,n]/cBNR[2,n] from TAOLDA are obtained with multiple energydifference calculations (see Equations (2) to (4)).
Results and Discussion
Singlettriplet energy gap
In order to determine the ground state of lBNR[2,n]/cBNR[2,n] (n = 6–100), we perform calculations based on spinunrestricted TAOLDA to obtain the lowest singlet and lowest triplet states of lBNR[2,n]/cBNR[2,n], with the respective structures being fully optimized. Subsequently, we calculate the singlettriplet energy gap (i.e., ST gap)^{19,46,47,48,50,51,52,55,56} of lBNR[2,n]/cBNR[2,n] as
Here, E_{S} and E_{T} are the lowest singlet and lowest triplet energies, respectively, of lBNR[2,n]/cBNR[2,n].
As shown in Fig. 2, E_{ST} decreases with an oscillatory pattern as n increases. Nonetheless, the oscillations are damped, and eventually vanish with the increase of n. The E_{ST} value of cBNR[2,n] exhibits a more pronounced oscillatory pattern than that of lBNR[2,n] when n is small, and monotonically converges from above to the E_{ST} value of lBNR[2,n] when n is sufficiently large. Besides, for considerably large n (e.g., n > 30 for lBNR[2,n] or n > 60 for cBNR[2,n]), the E_{ST} values of lBNR[2,n] and cBNR[2,n] monotonically decrease with the increase of molecular size. Note also that cBNR[2,n] possesses a larger E_{ST} value than lBNR[2,n]. On the basis of our TAOLDA results, for all the systems investigated (n = 6–100), lBNR[2,n] and cBNR[2,n] have singlet ground states (see Table S1 in Supplementary Information). However, the reason for the oscillations appeared on the smaller lBNR[2,n] and cBNR[2,n] may not be obvious, and hence, it will be interesting to build a simple model to explain this fact in the near future.
Note that the lowest singlet energies of lBNR[2,n]/cBNR[2,n] obtained with spinrestricted and spinunrestricted calculations should be the same for the exact theory due to the symmetry constraint^{46,47,48,60}. Nonetheless, KSDFT with conventional XC functionals may not satisfy this condition, especially for systems with radical nature^{41,42,46,47,48,49,50,51,60}. Here, we examine if the symmetrybreaking effects occur by additionally performing spinrestricted TAOLDA calculations for the lowest singlet states of lBNR[2,n] and cBNR[2,n], with the respective structures being completely optimized. The difference between the spinrestricted and spinunrestricted energies, obtained with TAOLDA, for the lowest singlet state of lBNR[2,n]/cBNR[2,n] is essentially zero (i.e., within the numerical precision considered in the present work), showing that unphysical symmetrybreaking solutions are not generated by our spinunrestricted TAOLDA calculations.
Note that the singlettriplet energy gaps of molecules are essential to understand many chemical processes. For example, there has recently been great interest in incorporating the singletfission phenomenon in solar energy conversion due to the improved energy conversion efficiency. As the singlettriplet energy gaps and the energetics of the singlet fission are closely related, accurate prediction of the singlettriplet energy gaps of molecules is critically important^{61,62,63,64}. Besides, molecules with small singlettriplet energy gaps are expected to be useful for thermally activated delayed fluorescence (TADF) applications^{63,65}. Therefore, the singlettriplet energy gaps of lBNR[2,n] and cBNR[2,n] reported in this work may provide insight into the singletfission phenomenon and TADF applications, which can be helpful for solar energy applications.
Vertical ionization potential, vertical electron affinity, and fundamental gap
It is interesting to examine whether lBNR[2,n] and cBNR[2,n] are useful for photovoltaic applications. Spinunrestricted TAOLDA calculations are carried out, at the groundstate structure of lBNR[2,n]/cBNR[2,n], to determine the vertical ionization potential^{19,47,48,50,51,52,55,56}.
vertical electron affinity
and fundamental gap
Here, E_{tot}(neutral), E_{tot}(cation), and E_{tot}(anion) are the total energies of lBNR[2,n]/cBNR[2,n] in the neutral, cationic, and anionic states, respectively.
As the system size increases, IP_{v} (see Fig. 3(a)) generally monotonically decreases (with a slight oscillatory pattern only for the smaller lBNR[2,n] (n ≤ 10) or smaller cBNR[2,n] (n ≤ 30)), EA_{v} (see Fig. 3(b)) generally monotonically increases (with a slight oscillatory pattern only for the smaller lBNR[2,n] (n ≤ 10) or smaller cBNR[2,n] (n ≤ 30)), and E_{g} (see Fig. 3(c)) generally monotonically decreases (with a slight oscillatory pattern only for the smaller cBNR[2,n] (n ≤ 30)).
Note that cBNR[2,n] possesses a larger E_{g} value than lBNR[2,n]. Besides, the E_{g} values of lBNR[2,n] (n = 15–73) and cBNR[2,n] (n = 21–89) range from 1 eV to 3 eV, lying in the ideal region relevant to solar energy applications. Our theoretical results for IP_{v}, EA_{v}, and E_{g} (see Tables S2 and S3 in Supplementary Information) may guide further experimental studies on lBNR[2,n] and cBNR[2,n].
Symmetrized von Neumann entropy
In view of the smaller E_{ST} and E_{g} values, the larger lBNR[2,n]/cBNR[2,n] are expected to possess more pronounced radical character in their ground states than the shorter lBNR[2,n]/cBNR[2,n]. To provide a quantitative measure of the radical character of lBNR[2,n]/cBNR[2,n], spinunrestricted TAOLDA calculations are carried out, at the groundstate structure of lBNR[2,n]/cBNR[2,n], to obtain the symmetrized von Neumann entropy^{19,47,48,50,51,52,55,56,60}.
Here, f_{i,σ} (i.e., a value between 0 and 1) is the i^{th} σspin (i.e., αspin or βspin) orbital occupation number obtained with spinunrestricted TAOLDA, approximately yielding the i^{th} σspin natural orbital occupation number^{46,47,48,53}. For a system with nonradical nature ({f_{i,σ}} take values in the vicinity of 0 or 1), S_{vN} is rather small. However, for a system with radical nature ({f_{i,σ}} can differ greatly from either 0 or 1 for spinorbitals with noticeable fractional occupations (i.e., active spinorbitals), and take values in the vicinity of 0 or 1 for other spinorbitals), the corresponding S_{vN} can grow rapidly with the number of spinorbitals that possess noticeable fractional occupations.
With the increase of system size, S_{vN} (see Fig. 4) generally monotonically increases (with a slight oscillatory pattern only for the smaller lBNR[2,n] (n ≤ 20) or smaller cBNR[2,n] (n ≤ 50)), implying that the larger lBNR[2,n]/cBNR[2,n] should possess increasing polyradical character in their ground states (see Tables S2 and S3 in Supplementary Information).
Occupation numbers of active orbitals
To illustrate why S_{vN} increases with the molecular size, we plot the occupation numbers of active orbitals for the ground state of lBNR[2,n]/cBNR[2,n], obtained from spinrestricted TAOLDA. For lBNR[2,n]/cBNR[2,n], the highest occupied molecular orbital, which is the (N/2)^{th} orbital, is referred to as the HOMO, the lowest unoccupied molecular orbital, which is the (N/2 + 1)^{th} orbital, is referred to as the LUMO, and so on^{46,48,50,51,52,53}, where N is the number of electrons in lBNR[2,n]/cBNR[2,n].
As presented in Figs 5 and 6, the occupation numbers of active orbitals for the ground state of cBNR[2,n] are distinctively different from those for the ground state of lBNR[2,n]. With the increase of molecular size, there are more and more orbitals with an occupation number close to 1 (i.e., there are more and more spinorbitals with an occupation number close to 0.5), obviously supporting that the polyradical character of the ground states of lBNR[2,n] and cBNR[2,n] should increase with n.
It is interesting to note that cBNR[2,n] with n = 7, 9, 11, and 13 possess pronounced diradical character in their ground states. Besides, as the active orbital occupation numbers are close to either 0 (unoccupied) or 2 (doubly occupied), cBNR[2,n] with n = 6, 8, 10, 12, 16, and 20 are expected to possess nonradical character in their ground states, showing consistency with the other electronic properties (e.g., the larger E_{ST} values, larger E_{g} values, and smaller S_{vN} values) associated with these relatively stable molecules.
Visualization of active orbitals
Here, we investigate the visualization of the active orbitals (e.g., HOMO −2, HOMO −1, HOMO, LUMO, LUMO +1, and LUMO +2) for the ground states of a few illustrative lBNR[2,n]/cBNR[2,n] (e.g., n = 10, 30, and 60), obtained from spinrestricted TAOLDA. As shown, the active orbitals are delocalized along the length of lBNR[2,n] (see Figs 7–9) or the circumference of cBNR[2,n] (see Figs 10–12).
Note that electron delocalization is a phenomenon in which electrons in a molecule are not associated with specific atoms or bonds, but are spread out over many atoms or bonds. As delocalized electrons are distributed over a greater region of space (e.g., many atoms or bonds), the net energy of the molecule is lowered, yielding resonance stabilization. Therefore, electron delocalization is an energetically favorable process^{66}. Moreover, since materials with several delocalized electrons tend to be highly conductive^{67}, the delocalized electrons of the boron nanoribbons are expected to enable enhanced electrical conductivity.
Relative stability
As mentioned previously, cBNR[2,n] possesses larger E_{ST} and E_{g} values than lBNR[2,n], implying that cBNR[2,n] should be more stable than lBNR[2,n]. Here, we assess the relative stability of the two isomers, i.e., lBNR[2,n] and cBNR[2,n] by calculating the relative energy of lBNR[2,n] with respect to cBNR[2,n].
where E_{S}(lBNR) and E_{S}(cBNR) are the groundstate (i.e., the lowest singlet state) energies of lBNR[2,n] and cBNR[2,n], respectively, obtained from spinunrestricted TAOLDA.
As presented in Fig. 13, with the increase of molecular size, E_{rel} generally monotonically increases (with a slight oscillatory pattern only for the smaller n values (n ≤ 30)). Besides, cBNR[2,n] are indeed more stable than lBNR[2,n] for all the n values studied (see Table S4 in Supplementary Information), revealing the role of cyclic topology.
Conclusions
In summary, nanosystems with radical nature, which are typically beyond the reach of traditional computational methods, have been accessible due to recent advances in TAODFT. In this work, we have employed TAODFT to predict the electronic properties (e.g., E_{ST}, IP_{v}, EA_{v}, E_{g}, S_{vN}, active orbital occupation numbers, visualization of active orbitals, and relative stability) of lBNR[2,n] and cBNR[2,n], with n ranging from 6 to 100. Since the ground states of the larger lBNR[2,n] and cBNR[2,n] have been shown to exhibit polyradical character, calculations based on KSDFT with conventional XC functionals may not reliably predict their electronic properties, and calculations based on accurate MR computational approaches are computationally infeasible for the larger lBNR[2,n] and cBNR[2,n]. Therefore, adopting TAODFT in the present study is well justified.
On the basis of our TAODFT results, lBNR[2,n] and cBNR[2,n] have singlet ground states for all the n values investigated. The electronic properties of cBNR[2,n] exhibit more pronounced oscillatory patterns than those of lBNR[2,n] when n is small, and approach the respective properties of lBNR[2,n] when n is sufficiently large. It is interesting to note that cBNR[2,n] with n = 6, 8, 10, 12, 16, and 20 possess nonradical character, and cBNR[2,n] with n = 7, 9, 11, and 13 possess pronounced diradical character in their ground states. Besides, the larger lBNR[2,n]/cBNR[2,n], which have the smaller E_{ST} values, smaller E_{g} values, larger S_{vN} values, and more pronounced polyradical character, should exhibit stronger static correlation effects than the smaller lBNR[2,n]/cBNR[2,n]. In addition, the visualization of active orbitals has revealed that the active orbitals are delocalized along the length of lBNR[2,n] or the circumference of cBNR[2,n]. From the relative stability of the two isomers, cBNR[2,n] are more stable than lBNR[2,n] for all the n values studied, revealing the role of cyclic topology. Owing to their sizedependent electronic properties, lBNR[2,n] and cBNR[2,n] can be promising for nanoelectronics applications.
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Acknowledgements
This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST1072628M002005MY3), National Taiwan University (Grant Nos NTUCC107L892906; NTUCCP106R891706; NTUCDP105R7818), and the National Center for Theoretical Sciences of Taiwan.
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S.S. and J.D.C. designed the project. S.S. performed the calculations. S.S. and J.D.C. contributed to the data analysis and writing of the paper.
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Seenithurai, S., Chai, J. Electronic Properties of Linear and Cyclic Boron Nanoribbons from ThermallyAssistedOccupation Density Functional Theory. Sci Rep 9, 12139 (2019). https://doi.org/10.1038/s4159801948560z
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