Simultaneous measurement of thermal conductivity and specific heat in quasi-two-dimensional membranes using the 3 ω method

With a view to 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 3 ω 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 ( 2 ω ) . 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 due to the combination of the total thermal resistance and total specific heat. Fitting Z ( 2 ω ) 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 it to measurements of the thermal conductivity at zero frequency.