Polaronic effects in monolayer black phosphorus on polar substrates

Abstract

We investigate the effect of charge carrier interaction with surface optical phonons on the band properties of monolayer black phosphorus induced by polar substrates. We develop an analytical method based on the Lee-Low-Pines theory to calculate the spectrum of Fröhlich type continuum Hamiltonian in the long-wavelength limit. We examine the modification of a band gap and renormalization of effective masses due to the substrate-related polaronic effect. Our results show that an energy gap in supported monolayer black phosphorus is enlarged depending on a particular substrate and the interlayer distance, z. Among the substrate considered, the largest gap broadening at z = 2.5Å is observed for the Al2O3 substrate, which is found to be ∼ 50 meV. Carrier-phonon coupling also renormalizes the effective masses which is more pronounced along the zigzag direction. Anisotropy of the effective masses becomes stronger by the influence of the polaronic effect corresponding to direction-dependent carrier-phonon coupling. We conclude that substrate phonons have a non-negligible effect on the static band properties of monolayer black phosphorus, which may be further exploited in its experimental and theoretical studies.

Attractive polaron formed in doped nonchiral/chiral parabolic system within ladder approximation

“…with Π(ω + Ω, p) = Π(ω + Ω, p, q = 0), and S is the volume of the space where all the chemical potential around the polaron are taken into account. Now the center of mass is just p. That's also agree with the results of the Fröhlich polaron model [19,55] in long-wavelength limit which with the strong electron-phonon coupling as well as the observable optical excitations [20]. For simplicity, we further set µ ↑ = 0 and the Fermi frequency Ω = 1 (which is possible in the zero-temperature limit), then the polaron self-energy becomes Σ(p, ω) = T (p, ω)G 0 ψ (0, 0) where G 0 ψ (0, 0) = 1 The self-energy is shown in Fig.7(a).…”

“…However, in most cases the repulsive polaron is thermodynamically unstable in the strong interaction region even at low-temperature. While for the attractive polaron, whose eigenenergy is negative, it can be observed in the solid state systems [13,16,17,18,19,20] including the topological materials, through the, e.g., substrate-related polaronic effect. Besides, the effects of both the intrinsic and extrinsic (like the electric field-induced Rashba) spin-orbit coupling on the polaron have also been explored [16,17,18,19], and it is found that the spin-orbit coupling is also related to the polaron-molecule transition [21].…”

“…While for the attractive polaron, whose eigenenergy is negative, it can be observed in the solid state systems [13,16,17,18,19,20] including the topological materials, through the, e.g., substrate-related polaronic effect. Besides, the effects of both the intrinsic and extrinsic (like the electric field-induced Rashba) spin-orbit coupling on the polaron have also been explored [16,17,18,19], and it is found that the spin-orbit coupling is also related to the polaron-molecule transition [21]. The quasiparticle residue Z (spectral weight), as an important property of the polarons, which can be measured by the Rabi oscillations [22], is finite even at zero-energy state for the parabolic system or normal Dirac semimetals, but it vanishes at zero-energy state for the Weyl semimetals which with higher dispersion [23,24] (i.e., the multi-Weyl semimetal).…”

Electronic and optical properties of bilayer blue phosphorus

“…Among these, the monolayer black phosphorus, 2D puckered structure of phosphorus, which was also successfully fabricated in laboratory [8,9] and studied with several theoretical works [10][11][12][13][14][15][16]. Moreover, another 2D structure of phosphorus with A7 phase which is known as blue phosphorus, is confirmed to be as stable as 2D black phosphorus due to the absence of imaginary frequencies in phonon spectrum [17][18][19].…”

Anisotropic hybrid excitation modes in monolayer and double-layer phosphorene on polar substrates

“…22,24 In most currently available 2DMs, a sample lies on the top of a polar substrate such as h-BN, SiO 2 , SiC or Al 2 O 3 . [25][26][27][28][29][30][31] In such heterostructures, the polar optical phonon modes of the substrate are localized near the 2DMs-substrate interface and are coupled to the surface optical (SO) phonon modes of the polar substrates through the long-range Fröhlich interaction. As a result, the SO phonon could be considered as the dominant plasmon-SO phonon coupling source in 2DMs on polar substrates.…”