Electron-phonon properties, structural stability, and superconductivity of doped antimonene

Antimonene is a recently discovered two-dimensional semiconductor with exceptional environmental stability, high carrier mobility, and strong spin-orbit interactions. In combination with an electric field, the latter provides an additional degree of control over the material's properties because of induced spin splitting. Here, we report on a computational study of electron-phonon coupling and superconductivity in $n$- and $p$-doped antimonene, where we pay special attention on the effect of the perpendicular electric field. The range of accessible hole concentrations is significantly limited by the dynamical instability, associated with strong Fermi-surface nesting. At the same time, we find that in the case of electron-doping antimonene remains stable and can be turned into a state with strong electron-phonon coupling, with the mass enhancement factor $\lambda$ of up to 2.3 at realistic charge carrier concentrations. In this regime, antimonene is expected to be a superconductor with the critical temperature of $\approx 16$ K. Application of bias voltage leads to a considerable modification of the electronic structure, affecting the electron-phonon coupling in antimonene. While these effects are less obvious in the case of electron-doping, the field effect in hole-doped antimonene results in a considerable variation of the critical temperature, depending on bias voltage.

Figure 1

Crystal structure of hexagonal antimonene; red lines denote the hexagonal unit cell, used in calculations. Lattice parameter, buckling constant, and bond length, denoted as $a,d$, and $b$ respectively, are also given on the figure.

Figure 2

Electronic structure of antimonene. (a) Band structure plotted along high-symmetry directions of the Brillouin zone. (b) DOS in the relevant charge carrier concentration range. Horizontal lines on panels (a) and (b) mark the following charge carrier concentrations (from bottom to top): $N_{\text{h}}= 1\times {10}^{15}{\mathrm{cm}}^{-2}, N_{\text{h}}= 3.7\times {10}^{14}{\mathrm{cm}}^{-2}, N_{\text{e}}= 4\times {10}^{14}{\mathrm{cm}}^{-2}$, and $N_{\text{e}}= 1\times {10}^{15}{\mathrm{cm}}^{-2}$. The corresponding Fermi surface contours are given in subplots (c)–(f).

Figure 3

Phonon specturm of antimonene. (a) Phonon dispersion plotted along high-symmetry directions of the Brillouin zone. (b) Phonon DOS.

Figure 4

DOS [(a), (b)] and critical temperature [(c), (d)] dependencies on charge carrier concentration for antimonene. Vertical dotted lines mark considered charge carrier concentrations. Multiple points for single $N_{\text{h/e}}$ on panels (c) and (d) represent $T_{\text{c}}$ at different values of ${\mu }_{\text{c}}^{*}$ (0.1, 0.12,…,0.2) from top to bottom. Red symbols and lines on panels (c) and (d) mark $T_{\text{c}}$ values at ${\mu }_{\text{c}}^{*}=0.1$. Dashed lines on bottom panels serve as a guide to the eye.

Figure 5

Contributions to total electron-phonon coupling (${\lambda }_{\mathrm{p}}$) and critical temperature ($T_{\mathrm{c}}$) as functions of hole (a) and electron (b) concentrations. Gray pentagons denote critical temperatures at ${\mu }^{*}$ = 0.1. Gray shaded region represents variability of $T_{\text{c}}$ with respect to ${\mu }^{*}$. Red triangles denote total electron phonon coupling. Contributions to ${\lambda }_{\mathrm{p}}$ from acoustic and optical phonon modes are shown by green squares and blue circles, respectively. Dashed horizontal line corresponds to $\lambda =1.3$.

Figure 6

Electron-phonon coupling and nesting function of doped antimonene. Panels (a) and (c) show nesting function resolved in $\mathbf{q}$-space for $N_{\mathrm{h}}=3.7\times {10}^{14}{\mathrm{cm}}^{-2}$ and $N_{\mathrm{e}}=1\times {10}^{15}{\mathrm{cm}}^{-2}$, respectively. Heat maps are given on the same scale for clarity, whereas actual maximum of $\xi$ is marked by a yellow arrow in panel (c). Panel (b) shows the Fermi surface contour for $N_{\mathrm{h}}=3.7\times {10}^{14}{\mathrm{cm}}^{-2}$, where arrows mark two distinct electron scattering channels discussed in the text. In dashed frame, panels (d)–(g) show electron-phonon coupling contributions from different phonon modes resolved in $\mathbf{q}$-space. Green labels in panels (c) and (d) mark the Brillouin zone and high-symmetry directions.

Figure 7

Effect of applied bias voltage on the electronic structure of antimonene. (a) DOS at various values of bias voltage. (b) Comparison of the electronic band structure for $V_{\text{b}}$ = 1.0 V (red line) and $V_{\text{b}}$ = 0.0 V (gray line).

Figure 8

Critical temperature dependence on charge carrier concentration at bias voltage of 1 V. Vertical lines mark relevant hole (a) and electron (b) concentrations. Six points for a single $N_{\text{h/e}}$ represent different values of ${\mu }_{\text{c}}^{*}$ (0.1, 0.12,…,0.2). Blue solid line represents $T_{\text{c}}$ at ${\mu }_{\text{c}}^{*}=0.1$ and $V_{\text{b}}=0.0$ V, and is given for comparison. Dashed lines serve as guides to the eye.

Figure 9

Dependence of critical temperature and electron-phonon coupling in doped antimonene on the bias voltage. Filled and open symbols correspond to $n$- and $p$-doping cases: (a) maximal critical temperature $T_{\text{c}}^{\text{max}}$; (b) charge carrier concentration $N_{\text{e/h}}$ corresponding to $T_{\text{c}}^{\text{max}}$; (c) total electron-phonon coupling strength $\lambda$. Panels (d) and (e) show variation of critical temperature and electron-phonon coupling strength with bias voltage at fixed concentrations $N_{\text{h/e}}=2\times {10}^{14}{\mathrm{cm}}^{-2}$.

Figure 10

Phonon spectra of hole- and electron-doped antimonene (left and right columns respectively) in $\mathrm{\Gamma }-M$ high-symmetry direction at various charge carrier concentrations, simulated using jellium doping technique. Dashed line corresponds to phonon spectra in undoped case.