Origin of different piezoelectric responses in elemental Sb and Bi monolayers

Sb and Bi monolayers, as single-element ferroelectric materials with similar atomic structure, hold intrinsic piezoelectricity theoretically, which makes them highly promising for applications in functional nanodevices such as sensors and actuators. Here, using density functional theory calculations, we systematically explore the piezoelectric response of Sb and Bi monolayers. Our findings reveal that Sb exhibits a negative piezoelectric response, whereas Bi displays a positive one. This discrepancy is attributed to the dominant role of different atomic internal distortions (internal-strain terms) in response to applied strain. Further electron-density distribution analysis reveals that the atomic bonding in Sb tends to be covalent, while the atomic bonding in Bi leans more towards ionic. Compared to the Sb monolayer, the Bi monolayer is distinguished by its more pronounced lone-pair orbitals electrons and associated larger Born effective charges. The Coulomb repulsions between lone-pair orbitals electrons and the chemical bonds lead to the Bi monolayer possessing more prominent atomic folds and, consequently, more significant atomic distortion in the $z$ direction under strain. These differences result in a considerable difference in internal-strain terms, ultimately leading to the reversed piezoelectric response between Sb and Bi monolayers. The present work provides valuable insights into the piezoelectric mechanism of two dimensional ferroelectric materials and their potential applications in nanoelectronic devices.

Figure 1

Crystal structure and electronic properties of Sb and Bi 2D monolayers. (a),(b) The crystal structure of Sb and Bi monolayers. Solid line boxes represent the unit cell of Sb and Bi. The red arrows indicate the intrinsic polarization direction. (c),(d) The electronic band structures of Sb and Bi monolayers. The size of red (blue) circles represents the contributions of the $p_z$ orbitals of ${\mathrm{Sb}}^{-}$ (${\mathrm{Sb}}^{+}$) and ${\mathrm{Bi}}^{-}$(${\mathrm{Bi}}^{+}$). The valance band maximum (VBM) and conduction band minimum (CBM) were identified as the end of green arrows. The energy difference of them was the band gap ($E_g$).

Figure 2

The polarization of Sb (a) and Bi (b) as a function of uniaxial strain. “$\varepsilon =0$” represents the ground states of Sb and Bi monolayers. The slopes of Sb/Bi-relaxed curves are the total piezoelectric coefficient ($e_{22}$) of Sb/Bi monolayers. The slopes of Sb/Bi-clamped curves are the clamped-ion terms ($e_{22}^0$) of Sb/Bi monolayers.

Figure 3

Changes of structural parameters of Sb and Bi under uniaxial strain. (a),(b) Crystal structure of Sb and Bi. The parameter $b$ is the lattice constant in the $y$ direction. $y_1$ and $y_2$ represent distances between adjacent atoms in the $y$ direction, and the sum of $y_1$ and $y_2$ is equal to half the lattice constant $b$ ($y_1+y_2=0.5b$). The parameter $z_1$ represents the distance between ${\mathrm{Sb}}^{-}$(${\mathrm{Bi}}^{-}$) and ${\mathrm{Sb}}^{+}$(${\mathrm{Bi}}^{+}$) atoms in the $z$ direction. (c),(d) Changes of fully relaxed structural parameters of Sb and Bi. (e),(f) Changes of structural parameters of Sb and Bi during the internal-strain process.

Figure 4

Bonding analysis of Sb (a) and Bi (b) monolayers. The density of states of ${\mathrm{Sb}}^{+}$ (${\mathrm{Bi}}^{+}$) and ${\mathrm{Sb}}^{-}$ (${\mathrm{Bi}}^{-}$) versus $E-E_f$ are shown in light blue, where $E_f$ denotes Fermi level. The red boxes highlight the difference in DOS between the two atoms near the valence band maximum, and the electron-density distribution (isosurfaces $=0.0007e/\mathrm{boh}{\mathrm{r}}^3$) in this energy range are visualized in the bottom subfigures.