Coupling between vibration and Luttinger liquid in mechanical nanowires

Abstract

The vibration of the mechanical nanowire coupled to photons via photon pressure and coupled to charges via the capacity has been widely explored in experiments in the past decades. This system is electrically neutral, thus its coupling to the other degrees of freedom is always challenging. Here, we show that the vibration can slightly change the nanowire length and the associated Fermi velocity, which leads to coupling between vibration and Luttinger liquid. We consider the transverse and longitudinal vibrations of the nanowires, showing that the transverse vibration is much more significant than the longitudinal vibration, which can be measured through the sizable frequency shift. We predict an instability of the vibration induced by this coupling when the frequency becomes negative at a critical temperature for the transverse vibrations in nanowires with low Fermi energy, which can be reached by tuning the chemical potential and magnetic field. The time-dependent oscillation of the conductance, which directly measures the Luttinger parameter, can provide evidence for this coupling. Our theory offers a new mechanism for exploring the coupling between the vibration and the electronic excitations, which may lead to intriguing applications in cooling and controlling the mechanical oscillators with currents.

Figures (4)

• Figure 1: Physical model and the excitations in the Luttinger liquid. (a) and (b) correspond to the T-vibration and L-vibration along the mechanical nanowire. The shaded black box corresponds to the clamped ends, and $y$ represents the displacement amplitude. (c) Two major scattering channels near the Fermi points, where $g_2$ and $g_4$ are measured in units of $2\pi \nu_0$ (see Eq. \ref{['eq-Hdiag']}).
• Figure 2: Energy scales of the vibration, plasmons, and temperature. The typical materials are set in the same size $(3\,\mathrm{\mu m},10\,\mathrm{nm}, 15\,\mathrm{nm})$ (notice that in nanotube, diameter $d = 4$ nm is used). The plasmon frequencies are denoted by $\omega_e$. Here we set $\nu_\text{F} = 10^4$ m/s, thus $\omega_e\approx16$ GHz. This energy mismatch can not be decreased by increasing the system sizes.
• Figure 3: Frequency shifts in nanowires with various materials. In (a) and (c), the solid lines are numerical results based on Eq. \ref{['eq-heff']}, and the dashed lines are our approximated frequency shifts using Eqs. \ref{['eq-deltaomega1a']} - \ref{['eq-deltaomega2b']} for nanowires with the same sizes (100 $\mu$m, 10 nm, 15 nm). In this condition, $y_{ ZP}$ is estimated to be of the order of $10^{-12}$ m. In InAs, $m^* = 0.023m_\mathrm{e}$ and $0.41m_\mathrm{e}$ for electron and hole, respectively, with $n_c = 10^{15}$/cm$^3$vurgaftman_band_2001. The Fermi velocity $\nu_\text{F}$ for InAs (electron), InAs (hole) and $\mathrm{SiO_2}$ (hole) are $7.9\times 10^4$, $4.4\times 10^3$, and $3.1\times 10^3$ m/s, respectively; while in graphene, nanotube and Al, are $8\times 10^5$, $8\times 10^5$, $1.0\times 10^6$ m/s. (c) Number of plasmons in the lowest mode. (d) $TS/U$, where $S$ is the entropy and $U$ is the Hamiltonian energy, as a function of $T$. When $T$ is high enough, this ratio will approach the upper bound 2 due to the linear nature of the collective excitations.
• Figure 4: Tunability of $\nu_\text{F}$. (a) The band structure of InAs nanowire with spin-orbit coupling strength $\alpha\approx 40\,\mathrm{meV\cdot nm}$liang_strong_2012vurgaftman_band_2001 and magnetic field $B_z=2\,\mathrm{T}$. (b) The corresponding $\nu_\text{F}$ at the Fermi points controlled by chemical potential. Near the dashed lines, $\nu_\text{F}$ can be regarded as vanishing small.