Phononic Frequency Combs in Atomically Thin Nanoelectromechanical Resonators Via 1:1 and 2:1 Internal Resonances
S M Enamul Hoque Yousuf, Jaesung Lee, Steven W. Shaw, Philip X.‐L. Feng
Abstract
We report on the first experimental demonstrations of phononic frequency comb (PnFC) generation in atomically thin molybdenum disulfide (MoS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) nanoelectromechanical systems (NEMS) vibrating near ~20MHz and ~50MHz in the high frequency (HF) and very high frequency (VHF) bands. Frequency comb patterns are generated by tuning two resonance modes with gate voltage ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{\mathrm {g}})$ </tex-math></inline-formula> to satisfy 1:1 and 2:1 internal resonance conditions. In the 1:1 internal resonance condition, we drive the two modes of a four-layer (4L) MoS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NEMS resonator at an anti-crossing in the nonlinear regime, where efficient nonlinear energy transfer occurs between the two coupled modes. The frequency comb characteristics are tunable by varying the RF driving voltage ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{\mathrm {drv}})$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{\mathrm {g}}$ </tex-math></inline-formula> . We find a threshold of PnFC generation at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{\mathrm {drv}}=550$ </tex-math></inline-formula> mV and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{\mathrm {g}}=-6.4\text{V}$ </tex-math></inline-formula> with relatively wide comb teeth spacing ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{\mathrm {r}})$ </tex-math></inline-formula> around 2.44 to 2.65MHz. In the 2:1 internal resonance condition, pumping a single-layer (1L) MoS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NEMS resonator at a frequency twice that of the fundamental mode ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{1})$ </tex-math></inline-formula> enables mode coupling between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{1}$ </tex-math></inline-formula> and the mode near <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2f_{1}$ </tex-math></inline-formula> , and generates PnFC with tunable <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{\mathrm {r}}$ </tex-math></inline-formula> . At <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{\mathrm {g}}=10\text{V}$ </tex-math></inline-formula> , pump voltage <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{\mathrm {p}}=60$ </tex-math></inline-formula> mV, and pump frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{\mathrm {p}}=48$ </tex-math></inline-formula> MHz, we observe PnFC with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{\mathrm {r}}\sim 45$ </tex-math></inline-formula> kHz, which can be tuned by varying <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{\mathrm {p}}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{\mathrm {p}}$ </tex-math></inline-formula> . We also demonstrate extraordinarily strong parametric amplification and spectral linewidth narrowing effects in 1L MoS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NEMS resonator and achieve parametric gain as high as ~10,000 (80dB) and spectral linewidth narrowing factor of ~5000 with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{\mathrm {p}}=153$ </tex-math></inline-formula> mV. The PnFCs demonstrated and the findings herein will be valuable for applications such as improving the sensitivity of resonant sensors. [2023-0017]