1. Introduction

Bismuth, discovered by Basil Valentine in 1450, is identified as a single element, in the form of an indistinct crystal. It was found to be hard and brittle. It can also be found in minerals such as Bismuth oxide (Bi₂O₃) and bismuthinite (Bi₂S₃). It is commonly found in silver-white color with reddish-brown marks found on its surface. Metal casting production could be a common application of Bismuth. Its mixed compound, in the form of a salt, can be used as a medical treatment for indigestion problems [Yang and Sun (2007)].

In generating power and device cooling, [Chen et al. (2003); Tan et al. (2016)] Bismuth (Bi) or its related compounds have received wide applications, especially in electronic devices Cheng et al. [2016]. All-possible phases, honeycomb ($\alpha ,\beta ,\gamma ,\delta$ and $\epsilon$) and non-honeycomb ($\xi ,\eta ,\theta$ and $\tau$) in two-dimensional form have been explored Zhang et al. [2015]. The transition of a material from semi-metallic to semiconducting can be achieved with a lower dimension. An investigation has been carried out, by comparing the electronic properties of a bulk (rhombohedral, $R3m$) and $\beta$-monolayer Bi. As one can see, the electrical conductivity of the monolayer appears to be lower than bulk material. By tuning the band structure, the transport properties can be enhanced significantly. Lowering the material dimension plays a significant role in changing the energy bands, especially in bands from T to L symmetry points, near the Fermi level ($E_F$). The bands located in between $\mathrm{\Gamma }$–Z direction near $E_F$ also play an effective role, in changing its electronic properties. It has been found that the bandgap energy has been increased (raised conduction band (CB), lowered valence band (VB)) when a $\beta$-phase monolayer has been compared to a bulk Bi material. Their cleavage energy vs inter-atomic displacement plot explains the exfoliation process of the monolayer. They have also performed some phonon dispersion calculations, and it appears that both bulk and monolayer are found to be dynamically stable because the phonon bands are all found positive (non-imaginary) [Wu et al. (2019)]. In general, their theoretical data analyzed, and the conclusion drawn is in good agreement with other recent literature [Cheng et al. (2014); Liu et al. (2017); Zhang et al. (2017); Reis et al. (2017); Akturk et al. (2016); Guo et al. (2017); Wu et al. (2018)].

He et al. investigated the electronic properties of Bi thin films grown on Au(111). The growth technique deployed is known as Stranski–Krastanov growth. The successful fabrication of Bi film structured ($\surd 37\times \surd 37$) Kagome and ($p\times \surd 3$) superstructure has been analyzed with low-energy electron diffraction (LEED) and scanning tunneling microscopy (STM). This STM experiment has been explored with Quantech LT-STM head (Ultrascan LT-100), employing a custom-built multi-chamber. The VB spectra have been observed with the ultraviolet photoemission spectroscopy (UPS) method. The intensity has been plotted against binding energy, and the spectra become broader as the no. of monolayers deposited on Au(111) increases. He et al. also investigated the work function dependence of the monolayer Bi. There seems to be a rapid fall when the Bi coverage (ML) increases from 0 to 1. The work function starts to decrease slowly when the no. of ML increases to 3. The work function no longer reduces when ML exceeds 4. In addition, the XPS measurements are carried out to gain a better understanding of how the material grows and the interactions between Bi and Au(111) atoms. It appears that there are no interactions between the Bi and Au(111) atoms because there are no shifts between the f orbitals on Bi and Au during the material growth. The difference between the in-plane repulse forces on the Bi atoms and the out-of-plane attractive forces occurring between Bi and substrate results in a Bi film formed. There is a clear indication on a Bi cluster (island-shape) segregated on Au(111), based on (1) the Bi 4f orbital rises exponentially. (2) The Au 4f orbital remains unchanged during the growth process. Their DFT calculations show that the Bi atom isolated from the Au(111) (on the hcp hollow site) is found to be stable. In the case of Ag(111), when growing Bi on top, the Bi atoms will be incorporated into the topmost layer of Ag(111), forming a silver-related compound Ag₂Bi, instead of a two-dimensional, stable Bi thin film. This occurs when 0.36ML of Bi is grown on Ag(111). The physical properties of low-dimensional Bi attract the wide attention of other researchers and scientists such as Fermi wavelength and Hall coefficient [He et al. (2019)].

Having light operating in small dimensions can be an advantage in terms of optical resonances since the light wavelength exhibited inside the nanostructure is much smaller. This could be beneficial in energy harvest or light confinement [Barnes et al. (2003)]. The charge carrier density, magnetoresistance, and quantum confinement effect have become the unique physical properties of semi-metal Bi, with a size of tens of nanometers. The real part of the dielectric constant, when showing negative (${\epsilon }_1<0$), may indicate having a plasmonic response at an excited state, in showing free charge carrier density. The optical response from the dielectric constants’ spectrum could affect the electronic band structure. The optical and plasmonic properties of nano-bismuth have been mainly investigated by Tian et al. A broad spectrum has been observed on the real dielectric function plot ${\epsilon }_1(\omega )$, with phonon energy ranging from 0eV to 1eV. The energy measured in between 1eV and 4eV exhibits excited plasmonic response since the ${\epsilon }_1$ found are negative. The EM spectrum covered is presented as follows: 0–0.1eV indicates far IR, 0.1–1.8eV indicates mid-IR to short-IR, 1.8–3eV indicates visible color, and $>3$eV indicates UV. Also, they have also reported a broad spectrum found on the ${\epsilon }_2(\omega )$ in between the Mid-IR and UV regions. The real part ${\epsilon }_1(\omega )$ can be calculated directly from the imaginary part ${\epsilon }_2(\omega )$ using Kramers–Kronig relations [Tian and Toudert (2018)].

The spin-orbit coupling, resulting in surface symmetry loss, suffers a strong splitting on the surface bands. This physical mechanism makes the surface of bulk Bi-metallic. The splitting of the surface bands here can fully explain the electronic structure of the Bi surface. Creating a spin source or a spin filter from reduced $\mathrm{\Omega }$ losses in resulting a high (low) speed (power) is crucial, especially in applications such as quantum computing. All possible types of Bi surfaces have been investigated by Hofmann et al. This includes the rhombohedral and hexagonal structures. Their structure models have been described with lattice vectors $a_1$, $a_2$, $a_3$, and $b_1$, $b_2$, $b_3$ (e.g. for rhombohedral, the reciprocal vector is ${\mathrm{\text{mb}}}_1+{\mathrm{\text{nb}}}_2+{\mathrm{\text{ob}}}_3$). These vectors can be represented by the cartesian coordinates x, y and z [Hofmann (2006)].

2. Computational Methods

In the pursuit of examining the structural, electronic, optical and mechanical properties, Kohn–Sham-based Density Functional Theory (DFT) [Hohenberg and Kohn (1964); Kohn and Sham (1965)] remains a remarkable tool for atomic simulations. In the present study, the Cambridge Total Serial Energy Package (CASTEP) [Segall et al. (2002)] is utilized with the non-local Heyd–Scuseria–Ernzerhof (HSE03) as exchange-correlation (XC) potential [Heyd et al. (2003); Krukau et al. (2006)]. The core-electron interactions are dealt with by applying Norm-conserving pseudopotentials (NCPP) [Hamann et al. (1979)]. The EDFT/All Bands method is applied [Marzari and Vanderbilt (1997)]. Minimization of the total energy was obtained by the optimization of the lattice parameters and relaxation of the atomic positions by conjugate gradient routine, with the energy cutoff of 600eV and Monkhorst Pack (MP) k-point grid of $2\times 2\times 3$ [Monhorst and Pack (1976)]. These setting reaches the maximum capacity of the computer, employing up to 30 core processors. Geometry optimizations are applied to provide a stable structure, with interatomic forces less than 0.03eV/Å, max. residual stress of 0.05GPa and total energy convergence of $1\times 10^{-6}$eV/atom using GGA calculations. The valence electron configurations for Bi are taken as $6s^{2}6p^{3}5d^{10}$.

3. Results and Analysis

In this paper, we perform a hybrid HSE03 in DFT a method of study on a carbon nanotube like Bi. The geometry optimization, band structure, density of states, dielectric functions and elastic constants are investigated. The Computational details are presented in Sec. 2. The results are discussed in Sec. 3. Finally, we conclude our work in Sec. 4.

3.1. Structural

Before any simulations can be made (except we have a Vacuum spacing of 15Å along the c-direction to avoid self-interaction under periodic boundary conditions), it is essential to find the optimized structures. A tetragonal (space group I4/MCM, No. 140) lattice structure has been deployed. This is shown in Fig. 1. After running the geometry optimization simulation. The optimized lattice constants are found to be $a=b=8.59$Å, and $c=4.14$Å, with bond angles of $\alpha =\beta =\gamma =90^{\circ }$. The atoms arranged in a unit cell are represented by means of charge density in Fig. 4. The fractional coordinates of Bi is shown in Table 2.

3.2. Electronic

The hybrid functionals suggested by Heyd, Scuseria, and Ernzerhof (HSE) in treating the core-electron interactions were adopted to be a better examination of electronic bandgap energy. The valence band top (VBT, located at A–M) is found to be higher than the conduction band bottom (CBB, located at A). This confirms the metallic property of Bi. We have also expanded Fig. 2(a) and managed to observe the CB and VB separation found on A and A–M symmetry points. They are found to be 0.217 and 0.296eV, respectively. The non-zero density of states found on the PDOS plot (around Fermi level) observed also verified this. The conduction and VBs located between $-5$eV and 5eV mainly correspond to 6$p^3$ orbitals. Other VBs observed from −15eV to −7.5eV contribute to the 6$s^2$ electron states. The transition $5d^{10}$ states make up the core levels, located from −22.5eV to −23.25eV.

3.3. Optical

The optical response may be one of the main interests as it exhibits some meaningful features of the electronic band structure. As one can see, the x component describes the material responses when the electric field is parallel to the interface (i.e. the incident light is perpendicular to the interface). Such response is much smaller in z-direction (i.e. the electric field is $90^{\circ }$ to the interface). So, we can neglect plotting the dielectric constants and corresponding quantities in z-direction. Considering the geometry of the studied material, the x-direction is the same as in the y-direction. We can consider plotting the polarization in x-direction only. The optical properties of the material can be understood by studying the optoelectronic behavior of the material.

The optical properties of Bismuth have been investigated widely [Toudert et al. (2017)]. There is an early literature published by Mallon and Runciman, investigating the optical properties of Bismuth, and they have reported the reflectivity of vibrations in the axial plane with a value of 56.3% at a wavelength of 486nm (phonon energy of 2.5eV). This is within the visible-light range (yellowish-orange). Such could be suitable in applications such as interband plasmonic and photocatalysis [Mallon and Runciman (1960)].

The imaginary part of the dielectric function ${\epsilon }_2(\omega )$ (in long wavelength limit) can be determined by solving the density of states and optical matrix elements together. This analyses the electronic state density and their interactions with EM radiation. This provides insight into the light-to-material response at longer wavelengths [Ambrosch-Draxl and Sofo (2006)].

Then, the real part of the dielectric function ${\epsilon }_1(\omega )$ can be derived directly from the imaginary part ${\epsilon }_2(\omega )$ by using the Kramers–Kronig relation [Kronig and Kramers (1928)].

Here, $|u_{k,i}〉$ and $|u_{k,f}〉$ are defined as the occupied initial and unoccupied final state with energy $E_{k,i}$ and $E_{k,f}$ with wave-vector k, $f(E_k)$ is the Fermi–Dirac distribution function, p is the dipole momentum, $\mathrm{\Omega }$ is the volume of the unit cell and e is the charge of electron.

Next, other optical moduli can be obtained from ${\epsilon }_1(\omega )$ by using the following expressions [Peatross and Ware (2015)] :

The optical absorption can be found using Arbi’s, Ehrenreich, and Cohen’s relation [Arbi et al. (2012); Ehrenreich and Morrel (1959)]. Here, optical absorption describes how much incident light has been absorbed when transmitted through the object.

$$\alpha (\omega )=\sqrt{2}\omega {(\sqrt{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}-{\epsilon }_1(\omega ))}^{\frac{1}{2}},$$(3)

$$\sigma (\omega )=\frac{j\omega }{4\pi }[({\epsilon }_1(\omega )+{j\epsilon }_2(\omega ))-1],$$(4)

$$R(\omega )=\left|\frac{\sqrt{{\epsilon }_1(\omega )+j{\epsilon }_2(\omega )}-1}{\sqrt{{\epsilon }_1(\omega )+j{\epsilon }_2(\omega )}+1}\right|,$$(5)

$$\eta (\omega )={\left(\frac{\sqrt{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}+{\epsilon }_1(\omega )}{2}\right)}^{\frac{1}{2}},$$(6)

$$\kappa (\omega )={\left(\frac{\sqrt{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}-{\epsilon }_1(\omega )}{2}\right)}^{\frac{1}{2}},$$(7)

$$L(\omega )=\frac{{\epsilon }_2(\omega )}{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}.$$(8)

In Fig. 5(b), there is only a peak found on the imaginary part of the dielectric function plot ${\epsilon }_2(\omega )$, and it is located at 29.8eV. The in-state real dielectric function ${\epsilon }_1(\omega )$ is 47.9. A high k-dielectric value has been found. The ${\epsilon }_1(\omega )$ graph reduces to its bottom at 2.37eV. This plot remains negative for a while before rises, crosses zero at 9.9eV, and remains at zero. The turning point of ${\epsilon }_1(\omega )$ from positive to negative at 1.85eV is the transverse optical (TO) frequency, while the turning point from negative to positive at 9.90eV is the longitudinal optical (LO) frequency. The inter-band transitions are related to the peaks in imaginary parts of the dielectric function ${\epsilon }_2(\omega )$. We have also observed that the real dielectric function ${\epsilon }_1(\omega )$ remains negative, i.e. Bi exhibits metallic properties, and waves do not propagate within visible (1.84–3.24eV) and ultraviolet (3.24–9.9eV) regions. At this point, the Drude function is found to be negligible (i.e. ${\epsilon }_1(\omega )$ is entirely ruled by interband transitions). The ${\epsilon }_1(\omega )$ is found to be positive within far IR to short wave IR range ($<1.84$eV).

The peak of the optical absorption spectra $\alpha (\omega )$ exhibits a peak at 5.53eV, which falls within the VUV range. There is another peak found at a very high frequency of 27.4eV. The optical bandgap $E_{\mathrm{\text{opt}}}$, which can be found by measuring the onset value of optical absorption spectra, is found to be 257meV, which is very small. This is represented by Fig. 5(a).

The peaks of the real conductivity $\sigma (\omega )$ are located at 1.74, 24.9 and 27.1eV, respectively. The imaginary part of the complex conductivity Im $\sigma (\omega )$ reduces to its minimum at 0.95eV, then rises to its maximum at 5.51eV. The plot then slowly attenuates, crosses the zero and reduces to its bottom at 23.4eV. This graph rises again, crosses zero and reaches its maximum at 28.2eV. This is represented by Fig. 5(c).

The reflectivity $R(\omega )$ measures a portion of the light that bounces off the surface when the light of the incident hits the surface of an object. The reflectivity curve $R(\omega )$ drops rapidly to zero after the energy of 9eV. The plot stays until 26.5eV, then the graph rises again and forms a small peak at 28.2eV. Up to 82% of reflections can be found at the highest peak, as shown in Fig. 5(d).

The in-state value of the refractive index is 6.93. The $\eta (\omega )$ plot rapidly falls to zero at 6.61eV, then the curve increases, and the refractive index remains low (0.6–1.1) within the range of 10.7 and 28.3eV. The peak of the extinction coefficient $\kappa (\omega )$ can be found at 1.87eV. When $\eta (\omega )$ and $\kappa (\omega )$ cross each other, the two intersection points are introduced to the TO and LO modes in the spectra. They are found at 1.87 and 9.98eV. This is shown in Fig. 5(e).

Finally, the loss of fast electrons (a collective oscillation of electrons in the VB) is reported to be 9.89eV. This is also the peak found at the EELS known as plasma frequency ${\omega }_p$. The real part of dielectric function ${\epsilon }_1(\omega )$ went through zero at this frequency. A small peak is also found at a high phonon energy of 28.2eV, as indicated in Fig. 5(f).

3.4. Elastic

A material response for macroscopic stress can be measured with elastic constants. The strength and mechanical stability of the solid and its bonding characteristics under different loading conditions can be provided with these computed quantities. In a tetragonal system, there are six independent constants, C${}_{11}$, C${}_{12}$, C${}_{13}$, C${}_{33}$, C${}_{44}$ and C${}_{66}$. The evaluated elastic constants showing positive values are shown in Table 3. We have compared our elastic data with theoretical work from Woodcox et al. [2020] Using second-order stress-strain relation [Kaurav et al. (2008); Singh et al. (2011)], the elastic constants C${}_{ij}$ can be evaluated using the following equation :

The elastic constants should satisfy the following criteria :

The tetragonal system is determined to be stable since all the above equations are satisfied. According to Fig. 3, C${}_{11}$ is less than C${}_{33}$, indicating the implied force on the a-axis is less rigid than the c-axis. C${}_{13}$ is slightly less than C${}_{12}$, which means the a-axis implied stress will not lead to a notable strain along the b-axis. Finally, C${}_{44}$ is found greater than C${}_{66}$, indicating the imposed bc-plane stress will produce a larger strain compared to one being composed on the ab-plane. We have compared our mechanical data with theoretical work from Woodcox et al. [2020].

3.5. Mechanical

The elastic moduli are evaluated using the relations given below:

The bulk and shear modulus can be found by simply averaging between the Reuss’s and Voigt’s scheme:

$$B=\sqrt{B_V{B}_R},$$(11e)

$$G=\sqrt{G_V{G}_R}.$$(11f)

The subscripts V and R are referred to as Voigt and Reuss, respectively [Voigt (1910); Reuß (1929)]. Meanwhile, B represents the bulk modulus and G represents the shear modulus.

$$E=\frac{9BG}{3B+G},$$(12)

$$\nu =\frac{3B-2G}{2(3B+G)}.$$(13)

Pugh’s ratio is defined as a portion of bulk modulus to shear modulus. The critical value of the Pugh’s ratio ($B∕G$) is 1.75. If the $B∕G$ value is less, the material is determined to be brittle. However, if the $B∕G$ value is higher, the material is determined to be ductile [Pugh (1954)]. In this work, the $B∕G$ value is evaluated to be 1.04, which is found to be brittle. About the Poisson’s ratio v, if Poisson’s ratio is less than 0.1, the material is found to be ionic. However, if Poisson’s ratio is within the range of 0.1–0.25, the material is found to be covalent [Haines et al. (2001)]. In this work, Poisson’s value v is calculated to be 0.14, suggesting bulk Bi to be covalent. The mechanical moduli of Bi are represented in Table 4.

4. Conclusion

The structural, electronic, optical, elastic, and mechanical properties of carbon-nanotube-like Bi material have been determined from first-principle calculations. In this work, we have identified the metallic nature of the Bi-material. Our first-principles calculations reveal that Bi are metal with CBB being lowered than VBT, with $E_g=-122$meV. The gap between the CB and the VB located on A (217meV) and A–M (296meV) symmetry points are also shown. The DOS of Bi was calculated, and it shows the CBs mainly correspond to the p orbitals, the VBs are made up of the s orbitals, and the transition d orbitals mainly contribute to the core levels. Our calculations also indicate that Bi is a material predicted with covalent and brittle properties. A high absorption coefficient of $1.89\times 10^5$cm${}^{-1}$ within the VUV range, a small optical bandgap measured to be around 257meV, and a high reflectivity of 82% indicating that it is more favorable for applications in optoelectronic devices. Finally, the calculation results in this work hopefully may give a contribution and additional contributions to reference and the literature for extended research on the more complicated structure of Bi.

Acknowledgment

The authors would like to thank colleagues and students from the Department of Physics in SUSTech for their helpful suggestions and inspiring discussions. This work is financially supported by the Shenzhen-Hong Kong Cooperation Zone for Technology and Innovation (GRANT NO. HZQB-KCZTB-2020050).

ORCID

Geoffrey Tse  https://orcid.org/0000-0002-3272-5175