Excitonic effects on the optical spectra of TiB₂ nanosheets

In this section, we have explained the theoretical model utilised to calculate the mechanical and elastic parameters. Generally stress–strain method is used to obtain elastic constants, where the stress response (σ) corresponding to strain (ε) can be modelled as

$$\begin{align} \sigma_i = \sum_{j = 1}^6 c_{i j} \varepsilon_j \end{align} \tag{ 1 }$$

The stress (σ) and strain (ε) are vectors having 6 independent components ($1\unicode{x2A7D} i,j \unicode{x2A7D}6$). C$_{ij}$ is second order elastic stiffness tensor expressed by a 6 × 6 symmetric matrix in units of GPa. The elastic stiffness tensor C$_{ij}$ can be determined based on the first-order derivative of the stress–strain curves. This method is, however quite computationally expensive, hence in the present study, elastic constants have been determined based on the energy–strain method, where energy variation corresponding to small applied strains to the equilibrium lattice configuration are recorded [40]. Under harmonic approximation, the elastic energy ($\Delta E\left(V,\left\{\varepsilon_i\right\}\right)$)of a solid is given by

$$\begin{align} \Delta E\left(V,\left\{\varepsilon_i\right\}\right) = E\left(V,\left\{\varepsilon_i\right\}\right)-E\left(V_0, 0\right) = \frac{V_0}{2} \sum_{i, j = 1}^6 C_{i j} \varepsilon_j \varepsilon_i \end{align} \tag{ 2 }$$

where $E\left(V,\left\{\varepsilon_i\right\}\right)$ and $E\left(V_0, 0\right)$ represents the total energies of distorted and equilibrium lattice cells, with volume V and V₀, respectively. In this method the elastic stiffness tensor (C_ij ) is obtained from second-order derivative of total energy versus strain curves [40]. Polynomial fitting procedure for calculating the second derivative at the equilibrium of the energy with respect to the strain is applied [41].

Laue classes of the hexagonal crystal system has the following form of the elastic matrix (C_ij ) with five independent elastic constants in its bulk form (space group 191, P6/mmm, hP1), where one of the element (C₆₆) can be expressed as combination of other independent elements ($C_{66} = \frac{C_{11} - C_{12}}{2}$) [42]

$$\begin{align*} C_{ij} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{bmatrix} \end{align*}$$

In 2D realm, only two independent elastic constants (C₁₁ and C₁₂) have been realised for hexagonal system, as the elastic matrix reduces to 3 × 3 form as follows [41]

$$\begin{align*} C_{ij}^\mathrm{2D hex} = \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{11} & 0 \\ 0 & 0 & \frac{C_{11}-C_{12}}{2} \end{bmatrix} \end{align*}$$

Born stability conditions [42] for hexagonal crystal system (bulk) have quadratic stability criteria as follows:

Criteria I: $C_{11} \gt |C_{12}|$

Criteria II: $2 C_{13}^2 \lt C_{33}(C_{11}+C_{12})$

Criteria III: $C_{44}\gt0$

Criteria IV: $C_{66}\gt0.$

However the criteria of mechanical stability in 2D realm [42–44] requires $C_{11}\gt0$ and $C_{11}\gt|C_{12}|$.

Further, the linearly varying elastic properties of 2D materials are mainly characterised by in-plane stiffness (C) and Poisson's ratio (ν), where C is determined by linear elastic constants and ν is the ratio between transverse contraction strain and longitudinal extension. For 2D hexagonal system, there are two different strain modes for which the strain vector (ε) is (δ, 0, 0) and (δ, δ, 0), respectively and the corresponding elastic energy ($\Delta E/V$) is $\frac{1}{2}C_{11}\delta^2$ and $(C_{11}+C_{12})\delta^2$, respectively. C₁₁, C₂₂, and C₁₂ represents the elastic stiffness along xx, yy, and xy directions, respectively. In 2D hexagonal nanosheets elastic stiffness along xx and yy directions is similar ($C_{11} = C_{22}$). So for 2D hexagonal nanosheets, in-plane stiffness C and Poisson's ratio ν can be expressed in terms of elastic strain tensor elements, C_ij as

$$\begin{align} C = \frac{C_{11}C_{22}-C_{12}^2}{C_{11}} = \frac{C_{11}^2-C_{12}^2}{C_{11}} \end{align} \tag{ 3 }$$

$$\begin{align} \nu = \frac{C_{12}}{C_{11}} \end{align} \tag{ 4 }$$

In this section we have detailed underlying theoretical methods to understand light–matter interaction and calculate optical properties of the 2D nanosheets. The linear optical properties can be obtained from frequency dependent complex dielectric function $\varepsilon(\omega)$ calculated in the framework of the independent-quasiparticle approximation [45], defined as:

$$\begin{align} \varepsilon\left(\omega\right) = \varepsilon_1\left(\omega\right)+\dot{\iota}\varepsilon_2\left(\omega\right) \end{align} \tag{ 5 }$$

where $\varepsilon_1(\omega)$ and $\varepsilon_2(\omega)$ are the real and imaginary parts of the dielectric function, and ω is the photon frequency. Frequency dependent imaginary part $\varepsilon_2(\omega)$ can be calculated from dielectric matrix which is calculated from the electronic ground states by a summation over empty states [46] as

$$\begin{align} \begin{aligned} \varepsilon_2\left(\omega\right) & = \frac{4 \pi^2 \mathrm{e}^2}{\Omega} \lim _{q \rightarrow 0} \frac{1}{q^2} \sum_{c, v, \mathbf{k}} 2 w_{\mathbf{k}} \delta\left(E_c-E_v-\omega\right)\\ & \quad \times |\langle c|\mathbf{e} \cdot \mathbf{q}| v\rangle|^2 \end{aligned} \end{align} \tag{ 6 }$$

where indices c and v corresponds to conduction and valence states and $\langle c|\mathbf{e} \cdot \mathbf{q}| v\rangle$ corresponds to the integrated optical transitions from valence to conduction states, while e and q are photon's polarisation direction and electron momentum operator, respectively. The integration is carried out on k by summation over k-points with a carried weight factor w_k . Also, frequency dependent real part $\varepsilon_1(\omega)$ can be calculated from the imaginary part $\varepsilon_2(\omega)$ by the Kramers–Kronig transformation [46] as

$$\begin{align} \varepsilon_1\left(\omega\right) = 1 + \frac{2}{\pi}\mathcal{P}\int_{0}^{\infty} \frac{\varepsilon_2\left(\omega^{{\prime}}\right)\omega^{{\prime}}}{\omega^{^{\prime} 2}-\omega^{2}+\dot{\iota}\eta} \,\mathrm{d}\omega^{{\prime}} \end{align} \tag{ 7 }$$

where $\mathcal{P}$ denotes the principal value and η is the complex shift parameter in Kramers–Kronig transformation which causes a slight smoothening of the real part of the dielectric function. Local field effects are however neglected in this approximation. Further from the real $\varepsilon_1(\omega)$ and imaginary $\varepsilon_2(\omega)$ parts of dielectric function, frequency dependent linear optical spectra of extinction coefficient $\kappa(\omega)$, absorption coefficient $\alpha(\omega)$, and reflectivity $R(\omega)$ can be obtained [47] as follows, where n is the refractive index

$$\begin{align} \kappa\left(\omega\right) = \left(\frac{\sqrt{\varepsilon_1^2 + \varepsilon_2^2}-\varepsilon_1}{2} \right)^{\frac{1}{2}} \end{align} \tag{ 8 }$$

$$\begin{align} \alpha\left(\omega\right) = \frac{\sqrt{2}\omega}{c}\left(\sqrt{\varepsilon_1^2 + \varepsilon_2^2}-\varepsilon_1\right)^{\frac{1}{2}} \end{align} \tag{ 9 }$$

$$\begin{align} R\left(\omega\right) = \frac{\left(n-1\right)^2+\kappa^2}{\left(n+1\right)^2+\kappa^2} \end{align} \tag{ 10 }$$

To take excitonic effects into account, we have employed the GW techniques to calculate the excited state properties. Based on GW-approximation, energies are calculated as

$$\begin{align} E^\mathrm{GW}_i = E^\mathrm{KS}_i + \Sigma_i \end{align} \tag{ 11 }$$

where $E_i^\mathrm{GW}$ and $E_i^\mathrm{KS}$ are GW quasiparticle energy and Kohn–Sham energy of the ith state, respectively, while $\Sigma_i$ is the self-energy of the ith state. Starting from standard DFT calculations for ground-state energy and electronic density, we calculated the linear optical spectra by solving BSE on the top of GW excited-state calculation (GW₀ approximation). The self-energy (equation (12)) in GW approximation [48, 49] is product of Green's function and dynamically screened interaction $W(\omega)$, ($W = \varepsilon^{-1}(\omega)\nu$), which is the bare Coulomb interaction ν screened by dynamical dielectric function $\varepsilon^{-1}(\omega)$

$$\begin{align} \Sigma^{GW}\left(r,r^{{\prime}},\omega\right) = \frac{\dot{\iota}}{2\pi}\int^\infty_{-\infty}\mathrm{d}\omega^{\prime} G\left(r,r^{{\prime}},\omega-\omega^{\prime}\right)W\left(r,r^{{\prime}},\omega^{\prime}\right) \end{align} \tag{ 12 }$$

Based on many-body quasiparticle framework [50, 51], GW formalism corrects the DFT based single electron eigenvalues by including exchange and correlation effects in self-energy term. The errors in determination of dynamically screened interaction $W(\omega)$ can be covered by including the solution of BSE performed on the top of GW calculations [52]. We have further utilised single-shot G₀W₀ approximation, where the screened Coulomb interaction W remains constant at its starting value while the one-electron Green's function G is self-consistently updated over the course of a single iteration. One could anticipate that the estimated optical characteristics via GW-BSE formalism will provide greater agreement with experiment.

The optical properties of 2D hexagonal layers are not properly treated using the macroscopic treatment as it incorporates the finite layer thickness (d = 0.3–0.7 nm for graphene [53]) in the limit of $d\rightarrow0$. For low-dimensional 2D systems, the dielectric function depends on the considered thickness of the vacuum region (L). In periodic stack of layers (multilayered system), sufficiently large interlayer distance has to be kept to avoid interactions between the periodic images of the 2D nanosheet [54, 55]. In order to circumvent such thickness problem, the L-independent optical conductivity of 2D layer $\sigma_\mathrm{2D}(\omega)$ can be used to characterise optical activity of any layered 2D system [56, 57]. For the optical activity of 2D layers, we are interested in the in-plane component $\varepsilon^\mathrm{in}(\omega)$ of the dielectric tensor, corresponding to light polarisation in a direction perpendicular to layer normal. For a freestanding 2D nanosheet (ML and BL) under the assumption of normal incidence, optical properties are mainly influenced by in-plane optical conductivity $\sigma(\omega)$. Further the in-plane component of macroscopic dielectric tensor can be related to optical conductivity $\sigma(\omega)$ [58] as,

$$\begin{align} \begin{aligned} \varepsilon^\mathrm{in}\left(\omega\right)& = 1+\frac{\dot{\iota}}{\varepsilon_0\omega}\sigma\left(\omega\right) \\ \sigma\left(\omega\right)& = \dot{\iota}\left[1-\varepsilon^\mathrm{in}\left(\omega\right)\right]\varepsilon_0\omega \end{aligned} \end{align} \tag{ 13 }$$

where ε₀ is the vacuum permittivity and ω is incident wave frequency. In such formalism if we change our artifical three-dimensional model (containing vacuum layer) to 2D nanosheet, the in-plane 2D optical conductivity $\sigma_\mathrm{2D}(\omega)$ [56, 59] relates as,

$$\begin{align} \begin{aligned} \sigma_\mathrm{2D}\left(\omega\right)& = L\sigma\left(\omega\right) \\ \tilde{\sigma}\left(\omega\right)& = \sigma_\mathrm{2D}\left(\omega\right)/\varepsilon_0c\\ & = \sqrt{\mu_0/\varepsilon_0}\sigma_\mathrm{2D}\left(\omega\right) \end{aligned} \end{align} \tag{ 14 }$$

where L is the slab thickness in the simulation cell and $\tilde{\sigma}(\omega)$ is normalised conductivity. Within independent-quasiparticle formalism, the dielectric function can be related to eigen states $|\nu\mathbf{k}\rangle$ and eigen values $\varepsilon_{\nu}(\mathbf{k})$ by Ehrenreich–Cohen formula [60], to describe real part of 2D optical conductivity as,

$$\begin{align} \begin{aligned} \operatorname{Re} \sigma_{2 \mathrm{D}}\left(\omega\right) & = \frac{8 \pi^2}{\varepsilon_0 \omega A} \sum_{\mathbf{k}} \sum_{c, v}\left|M_{c v}\left(\mathbf{k}\right)\right|^2 [\delta(\varepsilon_c\left(\mathbf{k}\right)\\ & \quad -\varepsilon_v\left(\mathbf{k}\right)-\hbar \omega)-\delta\left(\varepsilon_c\left(\mathbf{k}\right)-\varepsilon_v\left(\mathbf{k}\right)+\hbar \omega\right)] \end{aligned} \end{align} \tag{ 15 }$$

where A is the area of nanosheet and $M_{c v}(\mathbf{k})$ is electric-dipole matrix element for interband transition [46], defined as,

$$\begin{align} \begin{aligned} M_{c v}\left(\mathbf{k}\right)& = \lim _{\mathbf{q}_{\|} \rightarrow 0} \frac{e}{\sqrt{4 \pi \varepsilon_0}} \frac{1}{i} \frac{1}{\left|\mathbf{q}_{\|}\right|}\left\langle c \mathbf{k}\left|\mathrm{e}^{\mathrm{i} \mathbf{q}_{\|} \mathbf{x}_{\|}}\right| v \mathbf{k}\right\rangle \\ & = \frac{e}{\sqrt{4 \pi \varepsilon_0}} \frac{1}{i m \omega} p_{c v}^{\|}\left(\mathbf{k}\right) \end{aligned} \end{align} \tag{ 16 }$$

where $\mathbf{q}_{\|}$ is in-plane wave vector perpendicular to hexagonal axis, accounting for transitions between filled valence v and empty conduction c bands, and $p_{c v}^{\|}(\mathbf{k})$ is Bloch matrix element of in-plane component of momentum operator. In 2D nanosheets, reflectance generally has a small value and in the limit of vanishing reflectivity [60], the absorbance of the nanosheet can be related to 2D optical conductivity as,

$$\begin{align} A\left(\omega\right) = \frac{1}{\varepsilon_0 c} \operatorname{Re} \sigma_{2 \mathrm{D}}\left(\omega\right) = \sqrt{\frac{\mu_0}{\varepsilon_0}} \operatorname{Re} \sigma_{2 \mathrm{D}}\left(\omega\right) \end{align} \tag{ 17 }$$

where µ₀ is permeability of free space and c ($=\!\!1/\sqrt{\mu_0\varepsilon_0}$) is speed of light. Similar to the real part of the dielectric function, Kramers–Kronig relations can be utilised to derive imaginary part of 2D optical conductivity from real part in equation (15), as

$$\begin{align} \operatorname{Im} \sigma_{2 \mathrm{D}}\left(\omega\right) = -\mathcal{P} \int_{-\infty}^{+\infty} \frac{\mathrm{d} \omega^{{\prime}}}{\pi} \frac{\operatorname{Re} \sigma_{2 \mathrm{D}}\left(\omega^{{\prime}}\right)}{\omega^{{\prime}}-\omega}\\ \end{align} \tag{ 18 }$$

where $\mathcal{P}$ is cauchy principal value. For normal incidence, the optical properties of freestanding 2D layers in air, the reflectance R, transmittance T, and absorbance A independent of the light polarisation [56, 59] can be written as follows,

$$\begin{align} R = \left|\frac{\tilde{\sigma}\left(\omega\right)/2}{1+\tilde{\sigma}\left(\omega\right)/2} \right|^2 \end{align} \tag{ 19 }$$

$$\begin{align} T = \frac{1}{\left|1+\tilde{\sigma}\left(\omega\right)/2\right|^2} \end{align} \tag{ 20 }$$

$$\begin{align} A = \frac{\mathrm{Re}\tilde{\sigma}\left(\omega\right)}{\left|1+\tilde{\sigma}\left(\omega\right)/2\right|^2} \end{align} \tag{ 21 }$$

As the reflectance for 2D layers is extremely small, the absorbance (equation (21)) and the real part of the normalised conductivity (equation (17)) agree well and obtain similar values. Influence of frequency-dependent conductivity on optical properties of 2D layers is studied using above formalism. The dependency of the optical properties of 2D nanosheets on the band structure and electrical properties can be studied by calculating transition dipole moment (TDM), where the probability of an electronic transition is proportional to the square of the electronic TDM. For the transition between the initial a (valence) and final b (conduction) states in 2D nanosheets, the dipole transition matrix element which is the associated electric dipole moment, can be modelled as

$$\begin{align} \mathrm{P}_{a \rightarrow b} = \left\langle\psi_b|\mathbf{r}| \psi_a\right\rangle = \frac{i \hbar}{\left(E_b-E_a\right) m}\left\langle\psi_b|\mathbf{p}| \psi_a\right\rangle \end{align} \tag{ 22 }$$

where ψ_a and ψ_b are eigen states with E_a and E_b as their eigen energies andm is the electronic mass. The transition probabilities are calculated as the sum of the squares of TDM, P², in widely accepted units of Debye².

4.1.1. Structural model and stability.

4.1.2. Lattice, dynamical, and thermodynamical stability.

To study the lattice stability, we have plotted phonon dispersions along k-path M–Γ–K–M as shown in figures 1(d) and (e) for ML and BL, respectively. The absence of the imaginary frequencies reveals the dynamical stability of the 2D-TiB₂ nanosheets. The maximum frequency of the acoustic branch of BL (8.86 THz) is higher than that of the ML (8.54 THz) which can be possibly attributed to the smaller restoring force of atomic out-of-plane vibrations in ML. This can lead to the higher Debye temperatures and increased bond-stiffness in BL as compared to ML. In figure 2 we have compared the atom-projected phonon density of states (pDoS) for ML and BL. Due to the significant differences in atomic weights of Ti and B atoms, a large phonon frequency bandgap exists in 2D-TiB₂ nanosheets. We report the phonon frequency bandgap of 7.73 THz and 5.79 THz for ML and BL, respectively. The scattering between the acoustic and optical phonon modes can become forbidden with the increase in the phononic bandgap, which is evidently more pronounced in ML than in the BL. As evident from pDoS (figure 2), it can be concluded that the contributions to the acoustic modes in both ML and BL comes predominantly from Ti atoms, however, the optical modes are majorily contributed by the presence of the B atoms. To further study the thermodynamical stability we have performed the AIMD calculations at room temperature (300 K) for ML and BL (see supplementary material) and found energy fluctuations to be minimised indicating thermodynamical stability of 2D-TiB₂ nanosheets.

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4.1.3. Mechanical stability, properties and bonding Character.

The elastic properties of the 2D-TiB₂ nanosheets are discussed in this section. Mechanical and elastic parameters for 2D-TiB₂ nanosheets (ML and BL) calculated in applied strain range of $-1.5\%\unicode{x2A7D} \varepsilon \unicode{x2A7D}1.5\%$ are reported in table 2. ML and BL satisfies the stability criteria as defined in section 2.1. Eigenvalues ($\lambda_1, \lambda_2, \lambda_3$) of the stiffness matrix (in N m⁻¹) for ML are 97.027, 147.779, and 194.053 while for BL are 127.318, 254.636, and 399.882. The positive eigen values of stiffness matrix for both ML and BL indicates their stable nature. In table 2, we compared the independent elastic constants, modulii, and ratios for ML and BL. In-plane stiffness (Young's modulus E) for 2D layers provides a measure of rigidity under the application of external load. We observe that with an increase in single layer (from ML to BL), the in-plane stiffness almost doubles (see table 2). Graphene has the highest stiffness among 2D materials (340 ± 50 Nm⁻¹ [70]) owing to the presence of C–C bonds. The cumulative effect of presence of the additional layer (in BL) and slightly weakened bond strength due to the electron deficient nature of boron compared to carbon, explains the enhanced stiffness of 2D-TiB₂ BL, approaching the stiffness values for graphene [70]. Such nature of boron bonding can also possibly explain the reduced in-plane stiffness of ML in comparison to graphene. In-plane elastic parameters (elastic modulii E and G and Poisson's ratio ν_xy ) for 2D-TiB₂ nanosheets are highly isotropic in nature (see figure S1 in supplementary material). Poisson's ratio (ν) can be realised as a material's response towards the externally exerted stress on it. Mechanically stable 2D systems have ν between −1 and 0.5. Our calculated ν values for ML and BL (see table 2) falls in the range, indicating the mechanical stability of nanosheets.

Table 2. Independent elastic Stiffness Constants C_ij (in N m⁻¹), two-dimensional (2D) elastic modulii (Young's modulus E, Shear modulus G, and bulk modulus B, all in N m⁻¹ units), and elastic ratios (in-plane Poisson's ratio ν_xy and Pugh's $G/B$ ratio) for monolayer (ML) and bilayer (BL) 2D-TiB₂ nanosheets.

	C11 (N m−1)	C12 (N m−1)	E (N m−1)	G (N m−1)	B (N m−1)	νxy	$G/B$
ML	170.916	−23.137	167.784	97.027	73.890	−0.135	1.313
BL	327.259	72.623	311.143	127.318	199.941	0.222	0.637

The presence of the auxetic behaviour (negative Poisson's ratio NPR, see table 2) in ML is reported earlier [68]. Auxetic behaviour is generally observed in engineered bulk structures and is rare to sight in 2D materials [68, 71]. It is further shown that the addition of any extra layer (BL) destroys the auxetic behaviour (ν = 0.222) of freestanding 2D nanosheet. Such behaviour is mainly attributed to the presence of interlayer interactions in BL (and beyond) resulting in enhanced stiffness and non-auxetic behaviour. The increase in the Poisson's ratio (from negative to positive) can be fundamentally correlated to enhancement of in-plane stiffness in multilayer nanosheets. Distinct behaviour of Poisson's ratio for differently layered structures can also be attributed to the spatial constraints (in z-direction), where buckling height can be adjusted in multilayered systems while same cannot be realised in ML. Such behaviour is loosely based on atomic potential repulsion. The known 2D auxetic materials shows either in-plane NPR [72–79] or out of plane NPR [80–86], attributing to their re-entrant or hinged geometric structures [68, 87]. NPR cannot be attributed solely to geometry as both auxetic and non-auxetic behaviour is found in similar crystal structure for differently layered systems. Recently Yu et al showed that the auxetic behaviour of the materials could also be determined by their distinct electronic structures [87]. We have further investigated the distinct electronic structures of ML and BL (as discussed in section 4.2), which can possibly explain the origin of NPR in ML (in addition to structural features) and absence of auxetic behaviour beyond single layer of 2D-TiB₂. The strong coupling between d orbitals (Ti-atom) and p_z orbital (B-atoms) can possibly attribute for auxetic behaviour in ML [68].

Additionally, the brittle-ductile behaviour and bonding nature is studied by constraining $G/B$ ratio and in-plane poisson ratio ν_xy inline with the Pugh's [88] and Frantsevich [89] criteria. Transition between the brittle and ductile nature happens at $G/B = 0.57$ (Pugh's criteria [88]), where materials having $G/B\lt0.57$ behaves in a ductile manner while those with $G/B\gt0.57$ shows brittle nature. Our results (presented in table 2) indicates that both ML and BL are brittle in nature, however the brittleness has been drastically reduced by addition of a single layer. $G/B$ ratio for BL has remained less than half of the value for ML (table 2). Hence, such addition of multiple layers can be perceived as engineering parameter to change the behaviour from brittle to ductile nature. Also, constraining ν value at 0.26 (Frantsevich criteria [89]) provides additional insights into the brittle–ductile nature, where ν < 0.26 indicates brittle nature and for ν > 0.26 favours ductile nature. Based on constraining of ν value, both ML and BL are brittle in nature, while it is evidently seen in our calculations (table 2) that BL makes significant transition towards the ductile nature from the brittle behaviour of ML.

Further the bonding character of nanosheets is studied based on the Pettifor's criterion [90, 91], where positive (negative) Cauchy pressure indicates intrinsically ductile (brittle) nature attributed to non-directional metallic (directional covalent) bonds. Analogically to cubic crystals, in our present study for 2D isotropic hexagonal systems, we define Cauchy pressure as $CP = C_{12}-C_{66} = \frac{3C_{12}-C_{11}}{2}$. Hence we report CP for ML and BL as −120.164 and −54.695 N m⁻¹, respectively. It is evident that both ML and BL contains directional covalent bonds while being intrinsically brittle in nature where brittleness of BL is drastically reduced as compared to ML. The bonding character is also studied based on the electron localisation function (ELF) as discussed in section 4.2.

For application-based needs and experimental feasibility, free-standing ML needs to be stabilised on a substrate (h-BN [31], hexagonal MoS₂ [68]) to avoid its bending tendency [31, 92]. It is however recently shown that TiB₂ derived nanosheets can also act as cement additives to enhance stabilisation and hydration properties of C₂S [93].

Subsequently, we investigated electronic properties of ML and BL to study the origin of observed properties and interdependence of electronic structure on the optical properties. To extend the study of bonding character of nanosheets, we have calculated the ELF for ML and BL (see ELF maps for ML and BL at figure S2 in supplementary material). The B–B bonds are of covalent character owing to presence of localised electrons, however the strong localisation around small region in vicinity of metal atoms indicates the electron transfer from metal to electron deficient boron atoms, which helps in stabilising the nanosheets. ELF indicates that bonding in nanosheets is predominantly of covalent nature.

We further studied the band structure of the ML and BL using PBE functional and ultrasoft pseudopotentials along the K-path Γ–M $-K$–Γ. Dirac cone at the fermi level have been previously reported for ML [31], while we report the metallic kind of band structure for BL where few conduction and valence bands cross the fermi level. Band structure for both ML and BL are compared in figure 3, where contributions from atomic orbitals (Ti and B orbitals) is projected onto the band structure. Projected band structure suggests that near the fermi level major contributions are from Ti(d) and B(p) orbitals. Several dispersive (not flat) bands cross fermi level of BL, which are predominantly contributed by the Ti($d_{xz}, d_{yz}, d_{x^2-y^2}$) and B(p_z and traces of p_x ) orbitals. We further examined the effect of SOC on the band structures (see figures S3 and S4 in the supplementary material), to find that a bandgap of 0.057 eV is introduced in ML along with the small spin splitting of bands for both ML and BL. The in-plane (S_x and S_y ) contributions of spin is much pronounced compared to the z-component. We also plotted three-dimensional valence highest occupied molecular orbital (HOMO) and conduction lowest unoccupied molecular orbital (LUMO) bands (see figure 4) to study band contours and the dispersive nature. In figures 4(b) and (d), we have shown contour plots at the fermi level for ML and BL, respectively. The contour plot for BL clearly shows the anisotropic and dispersive nature of bands crossing the fermi level, while contour plot for ML shows the formation of dirac cones with anisotropic nature in agreement with previously reported findings [31]. In their recent work, Abidi and Koskinen [94] have studied the effect of different pure and hybrid exchange-correlation functionals for low-dimensional (specifically 2D) metallic systems. In their work they have optimised that HSE06 is the best and robust hybrid functional to work with 2D metallic systems followed by HSE03, PBE, and others. Hence to overcome the underestimations using PBE functional, we performed the band structure calculations of ML and BL TiB₂ nanosheets more accurately using hybrid HSE06 functional (see supplementary material). HSE06 indeed opens up a small amount of direct bandgap (0.115 eV) at Dirac point in ML, while the BL retains its metallic nature.

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We observe that the band nature changes from linear to dispersive with addition of single extra layer in ML, such that BL does not host any Dirac fermions with bands crossing the fermi level. To understand the changes in band nature, we probed ML by artificially varying the distance between the boron and titanium atoms starting from the relaxed parameter (h = 1.187 Å) with integral increment of 0.01 Å (see table 3). It is interesting to observe that with the increase in distance between titanium and boron layers, linear valence band crosses the fermi level (at titanium and boron interlayer distance of 1.32 Å, see table 3 and figure S5(k) in supplementary material) changing the nature of ML to metallic from a zero gap dirac material. It is evident from table 3 that upto the separation of 1.31 Å, dirac state in ML is nearly zero (narrow) gap with the gap in $\mathcal{O}(10^{-3})$ eV. Using parabolic fitting of the band edges, the effective mass of the charge carriers (holes and electrons in highest valence and lowest conduction band, respectively) is calculated till the bands cross the fermi level and is presented in table 3 for two distinct paths towards Γ and K point from the dirac point ($D = [0.14, 0.14, 0.00]$) of ML in the reciprocal space. It is clearly seen that D–Γ is the preferred direction of electron motion. Owing to the lower effective mass values for ML, it can be utilised for high-bandwidth applications [95]. Electronic structure also helps in unravelling the fundamentals for vanished auxetic behaviour with addition of single extra layer in ML (in case of BL). Further mechanical calculations for artificially achieved metallic state of ML reveals the NPR for metallic state, which clearly indicates that simply the crossing of fermi level by linear electronic band (metallic behaviour) does not destroy the auxetic behaviour. The presence of several bands crossing the fermi level in BL can be fundamentally correlated to the enhanced in-plane stiffness and hence positive Poisson's ratio.

Table 3. Electronic parameters of band structures along high-symmetry path Γ–M–K–Γ for artificially probed layer thickness h (in Å) for monolayer (ML) TiB₂. The system shows the metallic nature beyond the critical layer separation of 1.31 Å. D denotes the high-symmetry K-point [0.14, 0.14, 0.00] in the reciprocal space corresponding to the Dirac state of ML. $E_\mathrm{g}$ denotes the bandgap in units of eV, while $m^*_\mathrm{h}$ and $m^*_\mathrm{e}$ denotes the effective mass of hole and electron, respectively.

h	$E_\mathrm{g}$	$E_\mathrm{g}$	$m^*_\mathrm{h}$	$m^*_\mathrm{h}$	$m^*_\mathrm{e}$	$m^*_\mathrm{e}$
(Å)	Nature	(eV)	(D–Γ)	(D–K)	(D–Γ)	(D–K)
1.187	Direct	0.002	−0.020	−0.022	0.023	0.019
1.19	Direct	0.001	−0.020	−0.023	0.022	0.020
1.20	Direct	0.007	−0.022	−0.022	0.026	0.019
1.21	Direct	0.004	−0.020	−0.024	0.022	0.021
1.22	Direct	0.004	−0.021	−0.022	0.024	0.020
1.23	Direct	0.007	−0.020	−0.025	0.022	0.022
1.24	Direct	0.001	−0.021	−0.022	0.023	0.020
1.25	Direct	0.008	−0.024	−0.022	0.026	0.020
1.30	Direct	0.003	−0.023	−0.022	0.024	0.022
1.31	Direct	0.008	−0.027	−0.022	0.026	0.022
1.32	Metallic	—	—	—	—	—
1.33	Metallic	—	—	—	—	—
1.34	Metallic	—	—	—	—	—
1.35	Metallic	—	—	—	—	—
1.40	Metallic	—	—	—	—	—

We have further calculated the fermi surface for BL and constant energy isosurfaces for ML at different chemical potentials (µ) in the range of 0 to 0.5 eV for both valence and conduction bands to evaluate the fermi velocity. Development of ML branches from µ = −0.5 to 0.5 eV and BL for µ = 0 eV is shown in figure S7 of supplementary material, in the reduced zone scheme. We find that the average velocities across the fermi surface for both BL and ML are in the order of 10⁵ m s⁻¹. BL being intrinsically metallic in nature, shows several signatures of nesting which can perhaps leads to the charge density wave (CDW) formation. Experimental tuning of the distance between titanium and boron layers is quite challenging, attributing to the strong metal–boron bonds. It is recently shown that by disrupting the stable boron–boron ring structure, hydrogen storage capacity of similar class of 2D metal boride MgB₂ can be drastically increased [96], where partially exfoliated multilayers of MgB₂ were produced by mechanochemical exfoliation in zirconia resulting in 3–4 nm thick MgB₂ nanosheets [96]. Recent studies have developed various approaches to exfoliate TiB₂ into nanosheets, which opens up the avenue for practical applications such as enhancing the CO₂ capturing ability of CaO [36], enhancing the adsorption and activation of nitrogen [37], and as cement additives [93]. With such increased experimental footprints in budding novel direction for 2D metal borides, if we are able to control titanium and boron layers separation in TiB₂ nanosheets, by suggestive means of intercalations or other suitable experimental approaches, it would be promising to utilise 2D TiB₂ in device applications.

In 2D materials, excitonic effects plays an important role in correctly determining the optical activity, attributing to geometrical confinement and reduced Coulomb screening [97, 98]. To obtain reliable results, electron–hole interactions must be taken into consideration. In this last subsection, we will focus on the optical properties of the low-dimensional 2D nanosheets (ML and BL) of TiB₂. Due to the strong depolarisation effect in 2D planar geometry for light polarisation perpendicular to the plane, the optical properties for light polarisation parallel to the plane are discussed.

The imaginary $\varepsilon_2(\omega)$ part of the dielectric functions of ML and BL were calculated for the in-plane polarized directions as shown in figure 5. Both ML and BL shows better light absorption in NIR region, which is evident from the high absorption peaks (figure 5) at photon energies of approximately 0.16 eV (DFT), 0.82 eV (BSE) for ML and 0.12 eV (DFT), 1.01 eV (BSE) for BL. As shown in figure 5, small optical activity is also observed on the interface of IR and visible regions. Since majority component of natural light consists in visible and infrared regions, our findings suggest that such 2D materials are promising candidates for diverse optical, photocatalytic, and photovoltaic applications. The real $\varepsilon_1 (\omega)$ part of the dielectric function at zero energies provides the static dielectric constants, which were calculated to be 47.62 and 26.70 for ML and BL, respectively (see figure S8 supplementary material). Variation of $\varepsilon_1 (\omega)$ hints about the electronic polarisability of the nanosheets, indicating that with successive addition of layer, polarisability decreases.

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Real and imaginary parts of in-plane 2D optical conductivity for ML and BL (see figure S9 in supplementary material), provides insights about the optical response of the nanosheets in different regions (IR, visible, and UV). Based on the 2D optical conductivity, we further obtained optical spectra for ML and BL, at DFT (without the electron–hole interaction) and GW₀-BSE (excitonic effects included) levels. Our results of absorption, transmission, and reflection spectra as a function of the photon energy, as depicted in figure 6, reveals that inclusion of excitonic effects slight modify the spectral peak positions. For both ML and BL, the excitonic absorption edges are slightly blue-shifted, with relative absorption intensity increased for ML and decreased for BL. The reflection tendency is very less as compared to absorption and transmission values. In figure 6, reflection and transmission values are appropriately scaled for visualisation, where the transmission spectra can be seen as negative spectrum for absorbance values. We have marked the highest absorption peaks for ML (as P₁, P₂, and P₃) and BL (as 1, 2, *, 3, 4, 5, 6, 7, 8, 9, and 10) to observe the optical activity in different optical regions. As shown in figure 6, we find that including excitonic effects, ML show good ability to absorb light in the UV and NIR regions, which is evident from the high absorption peaks at approximately 0.86 (P₁), 3.67 (P₂), and 7.24 eV (P₃). It is interesting to note that with the excitonic effects, while the relative intensities of the peaks have reduced in BL, optical activity has been increased upon addition of an extra layer (in BL as compared to ML) across wide range of photon energies spanning IR, visible, and UV regions, which is evident from relative intensities and peak positions (as 1, 2, *, 3, 4, 5, 6, 7, 8, 9, and 10 in figure 6). Excitonic effects plays an important role in optical activity of 2D materials, which is evident from appearance of optical peak (*) in the visible region of BL (figure 6(d)), which can be regarded as the excitonic peak in BL. The optical activity across wide range of energies in BL can be attributed to the presence of several dispersive bands crossing the fermi level in BL as compared to the presence of linearly dispersive dirac cone in ML. Such distinct band nature of ML and BL near the fermi level is believed to be the reason for varied optical activity of differently layered nanosheets.

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To further examine the dependence of optical activity on the electronic structure, we have calculated the transition probabilities from the highest valence to the lowest conduction bands for ML. The transition probability for ML at the off-symmetry dirac point has high values of the order of 10⁴ Debye² (see figure 7), however including the SOC effects we observed that the transition probabilities span an area between the high symmetry M and K points in the order of 10⁸ Debye² because the energy is no longer strictly proportional to the wave vector. BL contains several bands crossing the fermi level, while we have calculated the transition probabilities between band indexes 10 and 11 (see figure S6 in supplementary material) to observe that transition peaks exists at multiple points in k-space, which is in agreement with the observed optical activity across wide range of photon energies. It is further interesting to observe that transition probabilities for BL are of the order of 10³ Debye², which is significantly less than ML, and even SOC does not change the order of transition probabilities of BL which is in agreement with our spectral results. The high transition probabilities of ML can be accounted to the presence of the linear bands and Dirac channels as very less to no threshold energy is utilised to move the carrier electrons from occupied to unoccupied states, which is further tunable by spintronic effects (such as SOC). Enhancement of transition probabilities with SOC can pave the way for spintronic applications. Finally, the enhanced optical activity of ML in infrared and ultraviolet regions can be attributed to splitting of linear bands regulated by spintronic effects which introduces small gap of 0.057 eV in ML. Further HSE06 calculations opens up direct gap (0.115 eV) at dirac point in ML accompanied by high transition probabilities, which can be attributed for sharp peaks in IR and UV regions. Also, the electronic polarisability of BL is reduced compared to ML which can be correlated with the decreased optical activity of BL. The metallic nature BL (at PBE and HSE06 levels) can also restricts its absorbance tendency which explains the reduced optical activity in NIR and UV regions. Being optically active in IR and UV regions, ML can be utilised for ultrafast photonic applications accounting to high values of transition probabilities, and BL being optically active in wide energy range can be utilised for device applications in broad regions of interest.

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