Ultracoherent self-assembled diamond nanomechanics reveals superfluid dynamics

Abstract

From gravitational-wave detection, protein force microscopy, to exploration of quantum-classical boundaries, many anticipated discoveries in fundamental science require improving measurement sensitivity limits. Through the fluctuation-dissipation theorem, mechanical dissipation sets the acoustic noise for this limit. Yet, even in high-purity crystals, the microscopic mechanisms responsible for the acoustic loss remain poorly understood. Tension-induced dissipation dilution offers a route to ultralow acoustic loss, but is challenging to implement in crystalline materials including single-crystal diamond. Here we realize a strain-engineered diamond nanomechanical platform using a liquid-assisted van der Waals self-assembly process that harnesses intrinsic surface forces to apply tensile stress exceeding 1 GPa. At cryogenic temperatures these resonators achieve quality factors beyond 10 billion (intrinsic material quality factors beyond 100 million). This exceptional coherence turns them into a sensitive probe for residual dissipation, elucidating three distinct two-level-system channels and one topological dissipation channel from a surface superfluid helium film. Our work shows how advancing mechanical coherence opens access to new regimes of physics in hybrid quantum systems, precision metrology, and condensed-matter physics.

Figures (16)

• Figure 1: Ultralow-loss diamond nanomechanics using van der Waals (vdW) self-assembly.(a) Schematic of the vdW self-assembly process: initially (i) suspended, parallel diamond nanobeams are drawn together by (ii) surface tension from liquid condensation. Upon evaporation, (iii) structural adhesion is achieved through vdW surface interactions. (b) False-color SEM image of segmented self-tensioning tethers enabling vdW self-assembly of complex string networks (illustrated above the SEM image) capable of supporting soft-clamped modes (two axes scaled differently). (c) Acoustic quality factor as a function of aspect ratio ($r=L/h$) for various nanomechanical platforms (<200nm thick) down to cryogenic temperature. Amorphous (red) and crystalline (blue) platforms are shown. Shaded regions indicate typical scaling behaviors from $\propto r$ (hard clamping) to $\propto r^2$ (soft clamping) Engelsen2024. Theoretical projections (1GPa) for diamond (100-nm thickness) at varying temperature highlight unprecedented performance achievable due to diamond's exceptional intrinsic quality factor (horizontal blue line). The estimated threshold for resolving topological dissipation in our experiment is shown as the dashed line. (d) Comparison of the projected force sensitivity with other mesoscale (red square) and atomic-scale (blue circles) platforms Sudhir25.
• Figure 2: van der Waals (vdW) adhesion physics.(a) Illustration showing how surface roughness reduces the effective vdW surface potential by averaging the vdW interaction across varying surface offsets. (b) Optical images of double-beam devices with uniform width (171nm), and (c) experimental characterization of vdW adhesion, correlating the length of the non-adhered region (labeled "split" in y-axis) to the beam separation gap. Comparison with FEM simulations (dashed lines) enables estimation of the effective vdW surface potential and the corresponding sidewall roughness between 0.3-0.6nm across devices. (d) Atomic force microscopy (AFM) surface topography maps of the top surface and sidewall of representative devices, with the sidewall roughness $R=0.5nm$ consistent with the previous analysis. (e) False-color SEM image of clamp-tapered vdW self-assembled beams (two axes scaled differently), and (f) the finite element (FEM) schematic (not to scale). (g) Measured average tensile stresses of optical-grade devices in various lengths (color-coded) as a function of clamp widths and gaps, closely matching FEM predictions using vdW surface potentials between $30-80mJ/m^2$. Discontinuities in simulated curves correspond to decohesion events between contacting surfaces (details in Supplementary Information).
• Figure 3: Self-assembled nanomechanical networks reveal two-level-system (TLS) defects.(a) Optical and scanning electron microscopy (SEM) images of a multi-tethered diamond perimeter-mode resonator with self-tensioning capability. (b) Comparison of mechanical quality factors measured at 5K for the first few soft-clamped (solid) and hard-clamped (open) modes in perimeter-mode resonators (red), alongside the fundamental modes of hierarchical resonators (solid green) and segmented clamp-tapered resonators (open blue). With the addition of some more devices, quality factors are measured at 12mK, where soft-clamped modes exhibit excess loss due to shear-coupled hydrogen defects. (c) Temperature-dependent measurements down to 5K of two hierarchical resonators (100-nm thick) from electronic and optical grade diamonds, matching well with the relaxational loss from a single defect family with an activation energy of $E_a = 35meV$ and tunneling rate of $\Delta_0 =9.1MHz$, identified as surface -OH rotors with a conceptual illustration stacked on top. Inset shows the averaged $Q_{\mathrm{int}}$ at 5K, showing surface-limited loss dynamics for electronic grade diamonds. (d) Conceptual illustrations of thermally activated defects in diamonds, including surface -OH rotors, hydrogen-induced and oxygen-induced subsurface defect complexes. (e) Temperature-dependent measurements of the first three modes of a perimeter mode resonator down to 12mK, showing distinct oscillatory behavior. Loss dynamics between 1-4K is removed from the transparent theory lines. Collective fitting to a distributed TLS model reveals the presence of three defect types (see inset for defect densities): a shear-strain sensitive TLS from hydrogen-induced defect complexes (green shaded), a normal-strain sensitive TLS from oxygen-induced defect complexes (red shaded), and a uniform background of amorphous TLS (gray shaded), all at the subsurface level (more details in Supplementary Information).
• Figure 4: Strain-engineered diamond nanomechanics reveals superfluid physics.(a) Conceptual illustration of vortex dynamics-induced damping from the 4He film on an infinite surface, where the vortex pairs switch between bound and unbound states during the topological phase transition. With an oscillating background superflow at the mechanical frequency $\Omega_m$, across the critical temperature $T_\mathrm{BKT}$, the characteristic vortex separation $d$ changes the vortex damping rate $\tau_D$, which leads to the maximized dissipation when $\Omega_m\tau_D\sim1$. (b) Measured excess damping induced by the topological phase transition, on the first three modes of a perimeter mode resonator, with amplitude and the shape consistent with a modified theory from the nanoscale effect of these resonator dimensions. The fitting incorporates both the vortex dynamics and a finite background contribution of TLS losses. (c) Measured frequency shift of the same resonator (undergone one temperature cycle) across the transition temperature at $T_c=1.42K$, characteristic of the superfluid thinning by the acoustic Casimir effect of surface modes. The fit serves as a guide to the eye, capturing the far-from-transition behavior of the Casimir physics. (d) FEM simulation of the out-of-plane mode of a beam structure ($L=200µm$, $h=200nm$, width $500nm$), and the surface oscillatory superflow in the resonator frame induced by the gradient inertial force.
• Figure A.1: (a) Optical image of a suspended hierarchical diamond string oscillator with up to six branchings, accompanied by (c) a false-color SEM image detailing the suspended nanomechanical strings within isotropically etched diamond trenches. (b) Finite element modeling (FEM) of the fundamental vibrational mode illustrating a soft-clamped displacement profile under zero tension, reducing radiation losses caused by clamp overhang by more than 100-fold. (d) Simplified fabrication flow starts with etching nanoscale features in a bulk diamond crystal using SiN hardmask. Next, atomic layer deposition (ALD) is used for sidewall protection, followed by quasi-isotropic oxygen plasma etching to undercut the structures, enabling the suspension of high-aspect-ratio slab-like nanomechanical string devices from single-crystal diamond plates. (e) Comparison of intrinsic quality factor $Q_{\mathrm{int}}$ for 100-nm-thick devices under different surface finishes--critical point drying, chlorine plasma, hydrogen plasma, and oxygen plasma--measured at room temperature (RT) and 5K for both optical grade and electronic grade diamond. Using X-ray photoelectron spectroscopy (XPS) spectra (Fig. \ref{['fig:SI_Fig1']}), we found that oxygen plasma treatment offers the optimal balance between plasma-induced damage and surface passivation, achieving the highest $Q_{\mathrm{int}}$ overall.