Specific heat measurement of thin suspended SiN membrane from 8 K to 300 K using the 3$\omega$-V$\ddot{o}$lklein method

We present a specific heat measurement technique adapted to thin or very thin suspended membranes from low temperature (8 K) to 300 K. The presented device allows the measurement of the heat capacity of a 70 ng silicon nitride membrane (50 or 100 nm thick), corresponding to a heat capacity of 1.4x10$^{-10}$ J/K at 8 K and 5.1x10$^{-8}$ J/K at 300 K. Measurements are performed using the 3$\omega$ method coupled to the V$\ddot{o}$lklein geometry. This configuration allows the measurement of both specific heat and thermal conductivity within the same experiment. A transducer (heater/thermometer) is used to create an oscillation of the heat flux on the membrane; the voltage oscillation appearing at the third harmonic which contains the thermal information is measured using a Wheatstone bridge set-up. The heat capacity measurement is performed by measuring the variation of the 3$\omega$ voltage over a wide frequency range and by fitting the experimental data using a thermal model adapted to the heat transfer across the membrane. The experimental data are compared to a regular Debye model; the specific heat exhibits features commonly seen for glasses at low temperature.