Simultaneous measurement of thermal conductivity and specific heat in quasi-2D membranes by 3ω thermal transport

TL;DR

The paper tackles the challenge of measuring thermal properties in atomically thin membranes by applying the 3$\omega$ thermal transport technique to suspended quasi-2D silicon nitride membranes. It develops a quasi-1D theoretical framework that relates the complex thermal impedance $Z(2\omega)$ to the membrane's thermal conductivity $\kappa$ and specific heat $c$, including parasitic channels such as wire conductance and gas load. Experimental data on SiN membranes show that fitting $Z(2\omega)$ across decades of frequency yields values of $\kappa$ and $c$ in line with literature, confirming the method’s validity and indicating negligible parasitic contributions under their conditions. The approach offers a versatile, single-wire calorimetric method to characterize thermal properties of ultrathin materials, with broad applicability to exfoliated 2D systems and combinations of low temperature, magnetic fields, and arbitrary sample environments.

Abstract

Toward measuring the thermal properties of exfoliated atomically thin materials, we demonstrate simultaneous measurements of the thermal conductivity and specific heat in suspended membranes. We use the 3ω technique applied to quasi-two-dimensional silicon nitride membranes having a metal line heater patterned on the surface to both deliver heat and directly measure the thermal impedance of the membrane at the heating frequency, Z(2ω). We derive an expression for the complex thermal impedance as a function of frequency, approximating the actual rectangular membranes with a one dimensional model. The derivation accounts for potential parasitic heat loss mechanisms including conduction along the heater line, and by the gas load in an imperfect vacuum. Qualitatively, the thermal impedance response resembles a low-pass filter, owing to the combination of the total thermal resistance and total specific heat. Fitting Z(2ω) to measurements across a few decades in frequency, we extract values of the thermal conductivity and specific heat of silicon nitride in agreement with literature values. We also study the dependence on the heating current, and compare to measurements of the thermal conductivity at zero frequency.

Figures (5)

• Figure 1: (a) Schematic of device: suspended rectangular membrane (orange, dimensions $l{\times}$w) on thermally conducting substrate (blue) with electrical connections and heating wire on membrane, (gold). Inset shows membrane thickness $d$ and heating wire dimensions $d_{\textrm{w}}$ and $b_{\textrm{w}}$. (b) Thermal impedance model including thermal conductivity and specific heat of the membrane and wire, and relevant thermal conductances $G_{int}$ between the wire and membrane; $G_{gas}$ to the cryostat thermal background; and $G_{sub}$ to the substrate. (c) Left: 2D model of membrane, with thermal boundary at the substrate temperature, and heating from a wire. Right: 1D model of a rod heated at the left by a thermal mass representing the wire, and cooled to thermal ground at right. (d) Real and imaginary parts of the thermal impedance at the heating frequency, $Z(2\omega)$. (e) In phase and quadrature data for $Z(2\omega)$ of a suspended Si$_3$N$_4$ membrane measured at $T = 200$ K, with fits to the $Z(2\omega)$ formula from which the membrane $\kappa$ and $c$ can be extracted.
• Figure 2: (a) Micrograph of SiN membrane device (100-nm-thick membrane appears orange). (b) In phase and quadrature $V(3\omega)$ voltages vs frequency at various temperatures. (c) Variation of $V(3\omega)$ with current at different frequencies, showing the expected $\propto I_{\omega}^3$ dependence. (d) Four-terminal resistance of the patterned heating wire vs temperature.
• Figure 3: (a) The thermal impedance at the heating frequency, $Z(2\omega)$, vs frequency and temperature. (b) Same as (a), with data fit to Eq. \ref{['zeq']}; 200 K is shown in Fig. \ref{['3w']}. Traces are vertically offset for clarity. (c) $\kappa$ and (d) $c$ for the SiN membrane extracted from the fits to $3\omega$ data (circles) and from separate measurements at dc (triangles; $\kappa$ only.). Error bars are symbol size or smaller. Comparison data are plotted by digitizing prior literature Lee1997Zink2004Queen2009Sikora2013Ftouni2013Ftouni2015.
• Figure 4: (a) Relative change in temperature, $\Delta T/T$, of the heating wire for various bias currents. (b) in phase and quadrature thermal impedance measured at different temperatures. (c) $\Delta T/T$ induced by a DC heating current. (d) thermal conductivity measured by DC method and AC method.
• Figure 5: Left: 2D model of Fig. \ref{['3w']}, with quasi-1D section highlighted. Right: quasi-1D model for calculating heat flow balance in a long uniform rod. Heat is injected by ohmic heating of the metal wire at left (by symmetry only half the membrane and wire is considered), and removed by thermal conductance $G_w$ out the ends of the wire, thermal transport $q_g$ by the presence of any background gas, and by flowing through the rod to thermal ground.