1 Introduction

Today, a fundamental challenge is a serious concern about finding renewable and clean energy resources because of ongoing decrease of non-renewable energy resources such as fossil fuels and their harmful effects on environment [1,2,3,4,5]. Therefore, myriad endeavors have been dedicated to development of energy storage devices and electrochemical energy conversion. Electronic devices, like electric vehicles, mobile phones [6], and laptops are increasingly using Li-ion batteries (LIBs) for rechargeable ion batteries [7], which are battery technologies of future [8,9,10,11] Nevertheless, it has been replaced by low-cost sodium-ion batteries (NIBs) due to scarcity of lithium and its rapidly increasing price [12,13,14,15] and also other metal ions battery such as Ca, Zn, Mg and etc. [16,17,18].

Recently, rechargeable ion batteries have been centered of considerable investigation efforts. Despite low specific capacity (372 mA h g−1), graphite is the most widely applied anode material for lithium-ion batteries that has good electrical conductivity and is low-cost and stable [19,20,21,22]. Since Na atoms have a larger atomic radius than Li, they have difficult intercalation into graphite. Thus, the use of graphite as the anode of NIBs faces limitations [23, 24]. Likewise, lithium–titanium-oxide (LTO) is another commercialized anode material that, despite its low capacity (175 mA h−1) and high cost, has remarkable thermal stability and cycle life [25]. The theoretical storage capacity of Si is up to 4200 mAhg−1, which is high and has been offered to replace graphite as an anode material. Nevertheless, delithiation/lithiation processes experience a great volume contraction/expansion (up to 300%), which cause structural degradation [26,27,28,29]. Hence, finding high-performance anode materials for NIBs have attracted attention in recent years from materials science perspective.

The interest in studying two-dimensional (2D) materials has been stimulated by discovery of graphene [30,31,32,33,34]. Due to confinement effects and attendance of electron–electron coupling, atomically thin 2D materials have delivered outstanding mechanical, optical, and electronic features [35,36,37,38]. Research groups have been motivated by extensive applications and fascinating attributes to explore for novel 2D materials for electrocatalytic reactions [39,40,41,42], solar energy conversion [43], photocatalysis [44,45,46,47,48], and energy storage [49,50,51,52,53]. The enhancing demand for next-generation NIBs can only be realized by developing performance of electrode materials. Cathode materials have improved faster than anode materials. Cost-effective graphite has favorable cycling stability and is extensively employed as anode material in most current NIBs [54]. However, low capacity of graphite, combined with its relatively weak interaction with Na atoms, limits its further application [55]. Poor bonding is a severe issue and endeavors were done to overwhelm this problem by forming defects on pristine graphene surface and introducing dopants. Unluckily, local structural alteration strategies affect slightly on improvement of anode material as they suffer mobility and stability challenges following Na adsorption, because of band gap opening [56,57,58,59,60,61]. As a result, next-generation NIB anodes should be made from 2D materials without structural distortions.

In present work, a novel binary compound (monolayer B3S) with relatively low symmetric honeycomb structure, flexible bonding ability, and remarkable electronic configuration has been proposed [62,63,64,65]. Via a global structural explore in conjunction with DFT computations, this novel 2D material with low sodium diffusion barrier and high theoretical capacity has been identified. Moreover, diffusion energy barriers for various feasible ways of sodium on B3S monolayer have been assessed employing nudged-elastic-band (NEB) method. Finally, for various adsorption concentrations of sodium on B3S monolayer, theoretical specific storage capacity and OCV have been computed and compared with formerly reported monolayers.

2 Computational method

2.1 DFT calculation

By applying B3LYP functional and 6-31G (d,p) basis set, electronic analysis, structural optimizations, and energy computations have been carried out. In order to anticipate weak interaction properly, dispersion term of Grimme “D” has been used [66]. According to literature review, B3LYP is an acceptable functional to compute structural and electronic features of various nanomaterials [67,68,69,70,71,72,73]. In present research, GAMESS software has been applied to execute all of the computations [74].

2.2 Calculation formula

By binding energy of Na atoms on B3S, adsorption strength can be determined. Binding energy (Eb) of adsorbed Na atoms on B3S is calculated as followings [75]:

$$ {\text{E}}_{{\text{b}}} { = }\frac{{{\text{(E}}_{{{\text{B}}_{{3}} {\text{S + Na}}}} - {\text{(E}}_{{{\text{B}}_{{3}} {\text{S}}}} {\text{ + nE}}_{{{\text{Na}}}} {))}}}{{\text{n}}} + {\text{E}}_{{{\text{BSSE}}}} $$
(1)

Herein, energy of an isolated Na in vacuum, energy of B3S, total energy of adsorbed Na atom on B3S, and total number of adsorbed Na atoms on B3S are represented by ENa, EB3S, EB3S+Na, and n, respectively. Basis set superposition error energy is represented by EBSSE, which is computed by counterpoise approach. OCV of Na-ions batteries anode materials is determined as below [17, 76, 77]:

$$ {\text{OCV = }} - \frac{{{\Delta G}}}{{{\Delta N}_{{\text{e}}} }} $$
(2)

Herein, electronic charge of Na ions, difference in number of Na ions at two different compositions in charge/discharge process, and difference in Gibbs free energy are represented by e (e = 1), ΔN, and ΔG, respectively.

$$ \Delta {\text{G }} = \, \Delta {\text{E }} + {\text{ P}}\Delta {\text{V }} - {\text{ T}}\Delta {\text{S}} $$
(3)

ΔG can be approximated to ΔE due to negligible contribution of entropy and volume terms [78]. Hence, following formula can be used to simplify OCV:

$$ {\text{OCV = }}{{\left( {{\text{E}}_{{{\text{B}}_{{3}} {\text{S}}}} {\text{ + nE}}_{Na} - {\text{ E}}_{{{\text{B}}_{{3}} {\text{S + Na}}}} } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{E}}_{{{\text{B}}_{{3}} {\text{S}}}} {\text{ + nE}}_{Na} - {\text{ E}}_{{{\text{B}}_{{3}} {\text{S + Na}}}} } \right)} {{\Delta n}_{{\text{e}}} }}} \right. \kern-0pt} {{\Delta n}_{{\text{e}}} }} $$
(4)

The Eb and OCV of Na atoms adsorbed on a B3S have been calculated as a function of sodium concentration. To model ion transport in batteries, researchers have used DFT to calculate the activation energy along diffusion pathways. The Modified Climbing Image Nudged Elastic Band (CI-NEB) method is often used to simulate diffusion kinetics. DFT has been employed to simulate the transport mechanism of ions in batteries by calculating the activation energy along the ion's diffusion paths [79]. The NEB method is a computational approach used in molecular dynamics simulations to study the motion of atoms along reaction pathways. It involves dividing the pathway into a series of discrete steps or images, optimizing the configuration of each image, and nudging neighboring images along the pathway to minimize the total energy of the system.

3 Results and discussion

3.1 Electronic features and Structural parameters of B3S monolayer

Following an extensive global structure search involving the binary combination of three Boron (B) and one Sulfur (S) as well as a comparison with previous research [80], we were able to identify three energetically favorable hexagonal B3S monolayers that displayed no buckling in the sheet. Nevertheless, it has been reported that the most stable B3S monolayer corresponds to the orthogonal primitive cell [81], as presented in Fig. 1a. The optimized structure of the B3S monolayer is shown in Fig. 1a. With lattice constants of a = 5.25 Å and b = 6.06 Å, B3S includes an orthogonal primitive cell (space group PMMA). Bond distances of boron–boron and boron–sulfur are between 1.662 and 1.678 Å and 1.815 to 1.849 Å, respectively. Also, B–B–B, B–B–S, and B–S–B bond angles are 125.8º, 114.7º, and 120.3º, respectively. Unlike graphene, these structural attributes cause B3S monolayer to be anisotropic. For B3S monolayer, isosurface plot of electron localization function (ELF) is shown in Fig. 1b, which red zone represents powerful covalent bonds are formed between boron–sulfur and boron–boron, as well as it has sp2 hybridization. It is interesting to note that value of ELF diagrams is around 0.5 in six-membered ring (hollow) zone, except for center of hollow, where there is not any electron distribution, which reveal that electrons in hollow will be unpaired and discrete. B3S monolayer hollow sites are therefore potential metal decoration sites. Because of valence band passing via Fermi level, B3S monolayer has metallic attributes. Also, the reported band gap value for B3S monolayer is about 0.74 eV, which agrees well with the value of 0.79 eV (seen Fig. 1S) obtained in our work [63, 82].

Fig.1
figure 1

a Geometrical optimization for the most stable B3S monolayer (top view) and b ELF of a sliced of the atomically thin structure B3S monolayer

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3.2 Binding energy of a single Na atom on B3S monolayer

Our focus was on applicability of B3S monolayer as an anode material for current NIBs due to its planar hexagonal graphene-like honeycomb structure with intrinsic metallic property. In computation of Na binding energy, 6 unique sodium adsorption areas were regarded: two bridge sites of boron-boron and boron-sulfur, two separate on-top sites of sulfur and boron atoms, and A1 and A2 inequivalent hollow sites, as indicated in Fig. 2. A1 and A2 sites have binding energies of − 1.34 eV and − 1.12 eV respectively, showing that Na atom prefers hollow sites. Vertical distances between B3S monolayer and Na at A1 and A2 sites are 1.67 Å and 1.89 Å, respectively. The energy values of absorption at B-sites were very weak, in addition, positive energy absorption values were obtained at T-sites. By p-orbital distribution, differences in binding length and binding energy of Na can be investigated. As hollow sites interact with the π-electron cloud, Na adsorption becomes stable (Table 1). In this case, π-electron cloud is predominantly formed by boron atom's pz orbital. There is a stronger interaction with the atom Na and higher π-electron cloud density at hollow sites with more boron atoms. As a result, A1 site may have better interaction with Na atom than the A2 site in terms of orbital interaction due to its more boron atoms. B3S monolayer delivers a relatively greater binding energy for sodium than foresaid anode materials (see Table 2), which indicate that structural stability of sodium is superior. By preventing metal clustering or dendrites forming on electrode surfaces, it can be more effective. Therefore, B3S monolayer can serve as an efficient anode material for Na-ion batteries.

Fig. 2
figure 2

Top view of Na adsorbed at a A2 and b A1 site of B3S monolayer

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Table 1 Computed value of biding energy (Eb) and vertical adsorption distance (d) of sodium atom from B3S monolayer at various feasible sites
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Table 2 Comparison of the OCV, diffusion barriers energy, and capacity of Na-ion on the different 2D-monolayers
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3.3 Diffusion attributes of Na ions on B3S monolayer

Across B3S monolayer, mobility of sodium ions impact on charge/discharge rate and plays a critical role in performance of battery. Hence, various feasible diffusion paths of sodium ions on B3S monolayer have been investigated and displayed in Fig. 3. A small sodium diffusion barrier is required to increase fast charge/discharge process. Diffusion barrier has been computed between closest adjacent inequivalent sites A1 and A2 employing nudged elastic band (NEB) approach. Three sodium diffusion paths (A1 → A2 → A1 → A1 indicated in Fig. 3) are obtainable from anticipated B3S monolayer. In order to describe applicability of electrode materials, vacancy and sodium diffusion barriers should be computed at high and low state of charge (SOC). In a low state of charge, calculated sodium diffusion barriers are 0.50, 0.48, and 0.41 eV for paths A1 → A2 → A1 → A1 (indicated in Fig. 3b) between hollow sites. Single vacancy at A2 is more stable than A1 when state of charge is high, which reveals Na prefers A1. Thus, vacancy diffusion path on one side of B3S monolayer must be A2 → A1 → A1 → A2, and related diffusion barriers are 0.81, 0.73, and 0.65 eV, respectively. In comparison to vacancy diffusion barrier, sodium diffusion barrier is low at a various state of charge. Moreover, vacancy and sodium diffusion barriers have been computed when contrary side of B3S monolayer is entirely sodium adsorbed. Comparable diffusion barriers reveal that attendance of sodium on other side of B3S monolayer has no effect on vacancy and sodium diffusions. For anode material purposes, Na and vacancy diffusions on B3S monolayer are comparable to those at Na diffusion barrier on graphite [83]. Hence, B3S may be more suitable anode material for NIBs.

Fig. 3
figure 3

a vacancy diffusion energy barrier at high state of charge and b energy barrier for Na diffusion on B3S monolayer at low state of charge

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3.4 Open-circuit voltage and storage capacity

In addition to factors such as diffusion energy barrier, electronic and structural attributes, open circuit voltage and storage capacity of B3S monolayer are also critical for efficient anode materials. Specific capacity is defined according to number of adsorbed atoms on surface of layered structure. By enhancing Na concentration, Na adsorption has been studied to assess fully Na adsorbed of B3S structure. The number of Na ions n used for this analysis was performed for n = 1, 2, 3, 4, 5, and 6. The B3S increases the specific capacity when a larger number of Na ions is adsorbed on the B3S matrix. At each Na concentration, all probable Na adsorption configurations were generated and maximum Na adsorption with exothermic negative Eads were estimated. With desirable Eads, maximum four Na atoms per B3S stoichiometric formula (Na4B3S) are adsorbable on two sides of B3S based on our computations. Unlike graphene, there are Na atoms in all hollow sites of B3S in both single-sided and dual-sided Na2B3S structures. Consequently, all hollow sites in Na-adsorbed systems can be filled by Na atoms because of larger lattice parameter in B3S and decreased Na–Na repulsion. NIB applications will therefore require high capacity of B3S. Then, specific capacity of the B3S structure for Na4B3S was assessed as below:

$$ Q = \frac{znF}{M} $$
(5)

Herein, Q is theoretical capacity, n is number of Na atom adsorbed, z is the number of ions in the electrochemical reaction, F is Faraday constant (96,485 C/mol), and M is molecular weight of B4S2 ring. A theoretical Na adsorption capacity of 1855 mA h g−1 has been computed for B3S monolayer, which is the highest reported value for two-dimensional monolayer anodes for NIBs. By comparing the results presented in Table 2, it can be concluded that the monolayer B3S has higher capacitance and lower OCV compared to other reported monolayers, making it suitable for sodium battery applications [84]. However, its diffusion barriers energy value is higher than some of the monolayers.

OCV of electrode materials also affects battery performance. NIB joints to cathodes require low open-circuit voltages on anodes in order to operate at their maximum voltages. Since open-circuit voltage is computed by chemical potential alteration of Na atom (μNa) based on SOC, it is essential to describe intermediate structures throughout charge/discharge process. At high and low states of charge, computed open-circuit voltage is 0.12 V and 1.53 V, respectively (see Fig. 4). Attendance of several Na atoms at high state of charge, causes repulsion between them and decreases attraction with monolayer. Hence, interactions between B3S monolayer and Na are weakened and voltage is decreased from low to high state of charge. OCV value for B3S (0.12 V) makes it a suitable option for NIB applications.

Fig. 4
figure 4

The OCV profile of Na adsorption on the B3S monolayer

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4 Conclusions

An investigation of B3S monolayer electrochemical, electronic, and structural attributes was carried out in present study in order to determine if it could be applied as anodes in NIBs based on first-principles DFT computations. Designed structure contains large lattice constants, which increase specific capacity with high Eads of Na. In addition, by applying DFT, Na diffusion barrier and OCV of B3S monolayer were computed. Based on unit formula of Na4B3S, theoretical storage capacity of B3S monolayer for NIBs reaches 1855 mA h g−1. Mean value of OCV is 1.53 V, which is less than conventional anode material and regarded as average value for anode materials. Obtained findings affirm that B3S monolayer is a promising and new anode material for NIB purposes. It can be expected that present work will provide a new motivation for further experimental and theoretical investigation on designed two-dimensional materials with acceptable absorption behavior.