  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.
