Gate Voltage Tunable Second Harmonic Generation in Mono- and Bi-layer Black Phosphene

TL;DR

This work tackles gate-voltage and chemical-potential control of second-harmonic generation in mono- and bilayer black phosphorus by combining a tight-binding description of electronic states with semiconductor Bloch dynamics under a static electric field. The authors derive and analyze SHG and EFISH conductivities, revealing layer- and direction-dependent nonlinear responses: monolayer BP shows strong armchair SHG with resonances near half the bandgap, while bilayer BP undergoes a topological semiconductor-to-semimetal transition forming Dirac cones, shifting the nonlinear response toward intraband, low-energy channels. Chemical potential further tunes inter- and intraband contributions, enabling precise, multi-parameter control of nonlinear signals. The results advance BP as a tunable nonlinear photonic platform and contribute to the broader field of topological nonlinear optics in anisotropic two-dimensional materials, with implications for modulators, frequency converters, and terahertz sources.

Abstract

Black phosphorene (BP) has emerged as a promising platform for tunable nonlinear photonics due to its layer-dependent bandgap, high carrier mobility, and remarkable in-plane anisotropy. This study investigates the second-harmonic generation (SHG) of monolayer and bilayer BP under an external static electric field, with describing the electronic states by a tight-binding model and the dynamics by semiconductor Bloch equations. Our results reveal that BP exhibits large second-order nonlinear optical response along the armchair direction, with significant resonant enhancement when the incident photon energy approaches half of its bandgap. Under an applied electric field of $10^7$ V/m, the effective second-order nonlinear susceptibility of BP can be as large as $10^3$ pm/V, surpassing that of the conventional nonlinear crystal AgGaSe$_2$ by more than an order of magnitude. With respect to the static electric field induced by gate voltage, we discuss the relation between the electric-field-induced second harmonic (EFISH) generation and conventional SHG -- under lower gate voltage, the EFISH approach agrees well with the SHG solutions, whereas the former is no longer applicable under higher gate voltage. Specifically, as the increasing gate voltage, monolayer BP exhibits the bandgap expansion and the corresponding blue-shift in the SHG resonant peak. In contrast, bilayer BP undergoes a semiconductor-to-semimetal transition, forming Dirac cone and generating divergent SHG spectra as photon energy goes to zero. Additionally, the chemical potential allows for precise control over interband and intraband nonlinear responses. This work provides important theoretical foundations for the development of BP-based tunable nonlinear photonic devices and expands the application potential of anisotropic two-dimensional materials in nonlinear optics.

Figures (7)

• Figure 1: Spectra of $3\sigma_0^{-1}|\sigma^{(3);dabc}|$ for monolayer BP with the static electric field direction along the (a) $x$-, (b) $y$-, and (c) $z$-directions, respectively. Here $\sigma_0 = \frac{e^2}{4 \hbar}$. Along each direction, the large elements are plotted with solid lines, specifically corresponding to $\sigma^{(3);xxxx}(\omega, \omega,0)$, $\sigma^{(3);xxyy}(\omega, \omega,0)$, $\sigma^{(3);xxzz}(\omega, \omega,0)$, and $\sigma^{(3);zxxz}(\omega, \omega,0)$; while other components are plotted by dashed lines scaled by a factor for visibility.
• Figure 2: (a) The band structure of monolayer BP for different gate voltage V. (b) The spectra of the conductivity $\sigma^{(2);xxz}(\omega, \omega)$ for monolayer BP for gate voltage $V=1,2,3$ eV. (c) Comparison between $\sigma^{(2);xxz}(\omega, \omega)$ for a gate voltage 0.01 eV and $3E_{\text{dc}}^c\sigma^{(3);xxzz}(\omega, \omega,0)$ with $E_{\text{dc}}^c= 4.7\times10^{7}$ m/V.
• Figure 3: The spectra of (a) $\sigma^{(3);xxxx}(\omega, \omega,0)$, (b) $\sigma^{(3);xxyy}(\omega, \omega,0)$, (c) $\sigma^{(3);xxzz}(\omega, \omega,0)$, and (d) $\sigma^{(2);xxz}(\omega, \omega)$ at gate voltage of $V=3$ eV, for different doping levels $\mu=\epsilon_{c\Gamma}+0.6$ eV, $\epsilon_{c\Gamma}+0.4$ eV, $\epsilon_{c\Gamma}+0.2$ eV, 0 eV, $\epsilon_{v\Gamma}-0.2$ eV, $\epsilon_{v\Gamma}-0.4$ eV, and $\epsilon_{v\Gamma}-0.6$ eV.
• Figure 4: (a) Band structure of bilayer BP. (b)-(d) Spectra of absolute EFISH conductivity $|\sigma^{(3)}(\omega, \omega,0)|$ for bilayer BP with 15 tensor elements. The largest elements under each gate voltage are shown with black solid lines, while the smaller components are indicated by dashed lines.
• Figure 5: Band structures of bilayer BP under vertical gate voltages with (a) $V = 1.0$ eV, (b) $V = 1.508$ eV, and (c) $V = 2.0$ eV. (d) Spectra of the $\sigma^{(2);xxz}(\omega, \omega)$ component under three representative vertical gate voltages. (e) The gate voltage dependent spectra of $\sigma^{(2);xxz}(\omega, \omega)$.