Mechanical and electrical properties of borophene and its band structure modulation via strain and electric fields: a first-principles study

Structural stability of borophene is the first issue discussed before further investigations. Herein, we performed the phonon dispersion calculation to confirm the structural stability of 2-Pmmn borophene. As shown in figure 2, the phonon spectrum and projected phonon density of state (PDOS) for in-plane optical branch B(X) and B(Y) as well as out-of-plane acoustic branch B(Z) were calculated by applying the PHONONPY code [38]. In the phonon spectrum of borophene, two small imaginary frequencies, one in near-gamma point and another in ${\rm{\Gamma }}$ -X, can be found, which indicates that 2-Pmmn borophene could not be dynamically stable against out-of-plane vibrations. This result is different from the previous study of monolayer boron sheet by Peng et al [2], but similar to the investigation of bilayer borophene by Zhong et al [27]. Since 2-Pmmn borophene has been synthesized, a reasonable explanation for this paradox is that the substrates could sustain borophene structure in a certain condition, such as low temperature and a small enough imaginary frequency value. As for projected PDOS, the B(Z) mainly contributes to the three low acoustic phonon branches, indicating that the whole borophene has a trend to translation along the Z direction because there is no constrain in this direction. Then, the B(X) mainly contributes to the high optical phonon branches, suggesting that the interaction between boron atoms in the X direction is stronger than the other two directions. Last, the vibrations in the Y direction are stronger than that in the Z direction but weaker than the X direction which can be ascribed to the buckled structure of borophene. Through the discussion of phonon spectrum and projected PDOS, we can conclude that the 2-Pmmn phase can be not the most energetically favorable structure among allotropes of borophene. However, on one hand, the free-standing 2-Pmmn borophene has not been studied clearly in experiments; on the other hand, for borophene on substrates, applications in microelectronics equally have a promising future. Besides, the phonon dispersions present an anisotropic behavior, which, in kinetics, provides the basis for mechanical and electronic properties.

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After discussing the structure stability, mechanical properties of borophene are further studied. Mechanical properties can be defined as the performance that materials behave when it is loaded by an external force. Meanwhile, the performance can be characterized by elastic stiffness constants. For orthogonal symmetry under plane stress, the four nonzero elastic stiffness constants can be written into the stress-strain formula as followed [39].

$$\begin{eqnarray}&&\left(\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\end{array}\right)=\left(\begin{array}{ccc}{C}_{11} & {C}_{12} & 0\\ {C}_{21} & {C}_{22} & 0\\ 0 & 0 & {C}_{44}\end{array}\right)\left(\begin{array}{c}{\varepsilon }_{xx}\\ {\varepsilon }_{yy}\\ {\varepsilon }_{xy}\end{array}\right)\end{eqnarray} \tag{ 1 }$$

where ${{\boldsymbol{\sigma }}}_{{\boldsymbol{ii}}}$ and ${{\boldsymbol{\varepsilon }}}_{{\boldsymbol{ii}}}$ are the stress and strain components, respectively. In our calculations, these elastic stiffness constants ${{\boldsymbol{C}}}_{11},$ ${{\boldsymbol{C}}}_{12},$ ${{\boldsymbol{C}}}_{22},$and ${{\boldsymbol{C}}}_{44}$ are 247.1, 4.08, 106.5, 525.5 GPa, respectively. The main engineering constants, Young's modulus E and Passion's ratio ${\boldsymbol{\upsilon }},$ can be derived as [40].

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{{\boldsymbol{x}}}=\displaystyle \frac{{{\boldsymbol{C}}}_{11}{{\boldsymbol{C}}}_{22}-{{\boldsymbol{C}}}_{12}^{2}}{{{\boldsymbol{C}}}_{22}}\,{{\boldsymbol{E}}}_{{\boldsymbol{y}}}=\displaystyle \frac{{{\boldsymbol{C}}}_{11}{{\boldsymbol{C}}}_{22}-{{\boldsymbol{C}}}_{12}^{2}}{{{\boldsymbol{C}}}_{11}}\,{{\boldsymbol{v}}}_{{\boldsymbol{xy}}}=\displaystyle \frac{{{\boldsymbol{C}}}_{12}}{{{\boldsymbol{C}}}_{22}}\,{{\boldsymbol{v}}}_{{\boldsymbol{yx}}}=\displaystyle \frac{{{\boldsymbol{C}}}_{12}}{{{\boldsymbol{C}}}_{11}}\end{eqnarray} \tag{ 2 }$$

where ${{\boldsymbol{E}}}_{{\boldsymbol{x}}}$ = 247 GPa, ${{\boldsymbol{E}}}_{{\boldsymbol{y}}}$ = 106.4 GPa, ${{\boldsymbol{v}}}_{{\boldsymbol{xy}}}$ = −0.038, ${{\boldsymbol{v}}}_{{\boldsymbol{yx}}}$ = −0.016 in this work.

Besides, according to the estimation formula for Young's modulus and Poisson's ratio, we calculated the constants for all angles. For the model of borophene in this work, which possesses orthogonal symmetry, the Young's modulus and Poisson's for all angles can be estimated by the formula followed [41, 42].

$$\begin{eqnarray*}&&Y\left(\theta \right)=\displaystyle \frac{{C}_{11}{C}_{22}-{{C}_{12}}^{2}}{{C}_{22}{\cos }^{4}\theta +A{\cos }^{2}\theta {\sin }^{2}\theta +{C}_{11}{\sin }^{4}\theta }\end{eqnarray*}$$

$$\begin{eqnarray*}&&\upsilon \left(\theta \right)=\displaystyle \frac{{C}_{12}{\cos }^{4}\theta -B{\cos }^{2}\theta {\sin }^{2}\theta +{C}_{12}{\sin }^{4}\theta }{{C}_{22}{\cos }^{4}\theta +A{\cos }^{2}\theta {\sin }^{2}\theta +{C}_{11}{\sin }^{4}\theta }\end{eqnarray*}$$

where the A = $\left({C}_{11}{C}_{22}-{{C}_{12}}^{2}\right)/{C}_{66}$−2 ${C}_{12},$ B = ${C}_{11}+{C}_{22}-\left({C}_{11}{C}_{22}-{{C}_{12}}^{2}\right)/{C}_{66}.$ In figure 3, the calculated angle-dependent Young's modulus and Poisson's ratio , which vary from 106.4 GPa to 305 GPa and from −0.816 to −0.017, respectively, suggesting the highly anisotropic mechanical properties of borophene. The negative Poisson's ratio is mainly owing to the quasi-planar structure. When the quasi-planar structure is stretched in one direction, and the corrugations are unfolded, leading to the expansion in all the in-plane directions.

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Furthermore, we investigated the strain-stress response by applied uniaxial and biaxial strain, respectively. Calculated biaxial and uniaxial stress-strain responses of borophene along the armchair and zigzag loading direction are illustrated in figure 4. In figure 4(a), the biaxial stress-strain curves for both armchair and zigzag directions include an elastic stage, yield stage and break stage. The lines gradually trend up to the ultimate tensile strength at which the material yields its maximum load-bearing ability. The critical stain could be easily found to be about 12%. By the linear fitting from 0 to 0.02 in the stress-strain curve, the Young's modulus along the armchair direction is estimated to be about 211 GPa, which is two times larger than that along the zigzag direction (84 GPa), indicating the anisotropic mechanical properties of borophene. Figures 4(c)–(d) shows the uniaxial stress-strain curves. Under uniaxial strain, the Young's moduli are 236 and 89 GPa along the armchair and zigzag directions, respectively, which is slightly larger than those under biaxial strain in both directions.

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To investigate the dynamical stability of strained borophene, we investigate the phonon spectrum of borophene with both tension and compression strains, as shown figure 5. For tension and compression strains smaller than 4%, no imaginary frequency is clearly identified, indicating the structural stability of borophene with relatively low strain. However, the tension and compression strains increase and become larger than 8%, obvious phonon imaginary frequency appears in Γ-X and Y-Γ. Therefore, the borophene remains its structural stability with applied strain lower than 4%, which becomes instable when the strain increases and exceeds 8%.

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The electronic band structure, the density of states (DOS) and project DOS (PDOS) of borophene are shown in figure 6. In figure 6(a), there are two bands cross the Fermi energy level in Γ-Y and M-X, together with DOS (figure 6(b)), in which forbidden band does not exist, indicating its metallic behavior along the armchair direction. On the contrary, band gaps, marked as BG1 and BG2, between the valence and conduction bands exist in Γ-Y and M-X. The anisotropy in electronic property indicates that borophene may behave as a semiconductor in this direction. The electronic properties of borophene are unusual and attractive.

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Borophene shows conductivity in the armchair direction, while semi-conductivity along the zigzag direction due to the existence of the band gap. The electronic band structure of borophene with applied uniaxial strain is shown in figure 7. With the increment of tensile strain, both BG1 and BG2 decrease slightly, whereas the overall properties remain unchanged.

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For the utilization in the future electronic application of borophene, its band structure regulation via applied strain and the electric fields was further studied to get a better knowledge of the electronic properties of borophene. The tendency of BG1 and BG2 under biaxial strain, armchair uniaxial strain and zigzag uniaxial strain ranging from −0.1 to 0.1, were detailed studied and shown in figure 8. For biaxial strain in figure 8(a), both the BG1 and BG2 decrease when biaxial strains increase. As for uniaxial strain, the bandgap has an orientation dependency. In figure 8(b), with the armchair uniaxial strain increasing, the BG2 decreases, and BG1 nearly remains still. In figure 8(c), zigzag uniaxial strains can lower the BG1 more obviously than BG2. On the other hand, in figure 8(d), when applying an electric field in the Z direction, ranging from 0 V/$\mathring{\rm{A}}$ to 0.5 V/$\mathring{\rm{A}} ,$ the BG1 decreases from 4.03 to 4.06 eV, while BG2 decreases from 9.03 to 9.05 eV. Since the range of change is very small compared to strain engineering, we could conclude that the applied electric field has almost no effect on the borophene band structure.

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As discussed above, the applied strain can be regarded as an effective method to regulate the electronic band structure of borophene. Here, the charge density in two specified cross-sections [010] and [−110], presenting the bonds along the armchair and zigzag directions, respectively, was analyzed to look into stain engineering under biaxial strain. In figures 9(a)–(c), the bonds were gradually elongated and the charge density in the center of the bond decreases, indicating the weaker B-B bonds. In figures 9(d)–(g), the charge density in the B-B bond decreases along the armchair direction. Thus, as the borophene is stretched, the bonds are weakened. This phenomenon can be attributed to corrugations in the zigzag direction. When the corrugations are unfolded, the localized bonds could be changed to a delocalized bond. On the whole, the bandgap can be decreased with the increased strain, and thus the enhancement of borophene conductance by strain engineering.

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