The optomechanical measurement of thermal transport in monolayer MoSe2 (Morell et al., Nano Letters 2019) yielded the thermal conductivity κ(T)\kappa(T) and specific heat C(T)C(T) of a suspended monolayer MoSe2 drum from the delayed photothermal (dynamical-backaction) response of its mechanical resonance, and left a precise, linked set of questions open: quantifying the thermal interface resistance between the monolayer and the substrate; extending the thermal time constant τ\tau, heat capacity, and thermal-conductivity measurements up to room temperature; probing the predicted phonon-hydrodynamics regime in monolayers, including access to the second-sound velocity, through τ\tau; and determining whether the size-dependent divergence of the thermal conductivity in two dimensions arises from dimensionality or from a population of long-lived ballistic phonons. These questions remain open in the literature: second sound and Poiseuille flow have been observed only in bulk graphite, the optomechanical methodology has been extended to few-layer (not monolayer) membranes, and the mechanism of the 2D size divergence remains actively debated.

Background

Two-dimensional (2D) crystals peeled or grown from layered van der Waals materials are one to a few atoms thick, and heat in them is carried almost entirely by lattice vibrations (phonons). Transition metal dichalcogenides (TMDs) such as MoSe2 are semiconducting 2D crystals whose monolayer form is central to 2D electronics and optoelectronics, so understanding how heat spreads within a monolayer — and escapes from it into the substrate — matters both fundamentally and for thermal management of devices.

Morell et al. (Nano Letters 2019) introduced an optomechanical way to measure this. A circular monolayer flake suspended over a cavity in a substrate forms a drum-like nanomechanical resonator. A power-modulated laser heats the drum center; because the drum's thermal expansion follows the heating with a delay, the resulting photothermal force drives the drum motion with a measurable phase lag (this delayed coupling is called dynamical backaction). From the frequency dependence of the response one extracts the thermal time constant τ\tau — the characteristic time for heat to flow from the illuminated spot through the monolayer to the supporting rim — and, with a heat-transport model, the in-plane thermal conductivity κ(T)\kappa(T) and the specific heat C(T)C(T) from cryogenic temperature upward. The measurements resolved the crossover from ballistic to diffusive phonon transport (in ballistic transport phonons cross the device without scattering, so the effective conductivity grows with device size; in diffusive transport κ\kappa is a size-independent material property) and agreed with first-principles calculations.

Several questions were left open. First, the thermal interface resistance — the additional thermal resistance at the contact between the monolayer and the substrate, analogous to a thermal boundary (Kapitza) resistance — was inferred to be small at the drum diameters used but was not quantified; smaller-diameter drums would magnify its relative contribution and allow it to be separated from the intrinsic in-plane phonon transport. Second, the technique operated only at low temperatures, because extracting τ\tau and CC up to room temperature requires a higher mechanical quality factor (a sharper resonance), obtainable for example with larger drums. Third, the technique was proposed as a route to the phonon-hydrodynamics regime, in which phonon-phonon collisions that conserve crystal momentum (normal scattering) dominate over momentum-relaxing processes (Umklapp and impurity scattering), so that heat flows like a viscous fluid; in this regime temperature disturbances propagate as a damped wave called second sound, and measuring τ\tau could give direct access to the second-sound velocity. Predictions for 2D crystals must also contend with flexural phonons — out-of-plane vibrational modes whose quadratic dispersion at long wavelength underlies several 2D-specific transport effects. Fourth, the technique was proposed as a tool to address the long-standing puzzle of anomalous heat conduction in two dimensions: in 2D systems with momentum-conserving dynamics the thermal conductivity is predicted to grow with system size (asymptotically like ln⁡L\ln L), and it is debated whether the size-dependent κ\kappa measured in suspended 2D crystals reflects this dimensional mechanism or simply a population of phonon states whose mean free paths are so long that they remain ballistic across the whole sample.

Problem Statement

Using and extending the optomechanical thermal-measurement technique demonstrated on suspended monolayer MoSe2 drums, produce a complete, temperature- and size-resolved, interface-separated quantitative account of phonon heat transport in suspended monolayers of two-dimensional materials. The account requires four linked, jointly supporting results:

(i) Quantify the thermal interface resistance between the suspended monolayer and the supporting substrate and disentangle it from the in-plane conductivity.

(ii) Extend the measurement of the thermal time constant τ\tau and of the heat capacity (and in-plane thermal conductivity) from cryogenic temperature up to room temperature.

(iii) In the temperature window where hydrodynamic phonon transport is predicted, determine whether a hydrodynamic (driftless or drifting) regime is observable in a suspended monolayer.

(iv) Use such temperature- and size-resolved measurements to help determine the origin of the size-dependent divergence of the in-plane thermal conductivity in two dimensions: whether it arises from the dimensional mechanism (long-wavelength momentum-conserving phonons) or from a population of anomalously long-lived ballistic phonon states.

These four components are jointly posed aspects of one common objective on one experimental platform: each is a measurable facet of the single question of how phonons carry heat in a two-dimensional monolayer and how that transport is shaped by the substrate interface, by hydrodynamic collective motion, and by finite system size. The related extension to anisotropic thermal conductivity in black phosphorus is not part of this problem, since it had already been demonstrated by Islam et al. (2018) prior to the source work.

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