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sinusoidal output voltage v$_{ac}$ with a frequency o f 1.26 MHz. Oscilloscope and spectrum analyser measurem ents of the signal are shown in Fig. 2b, 2c and 2d. The mo tion of the device is also studied by stroboscopic illumina tion under a microscope. The stroboscope is triggered by the electrical output v$_{ac}$ of the heat engine. By adjustin g the phase delay of the stroboscope a slow-motion mov ie of the engine’s sinusoidal motion is made (see Supple men- tary Video). Stroboscopic and electrical measure ments show that just above I$_{t}$ the displacement amplitu de of the mass depends very sensitively on the DC current I$_{dc}$ . If I$_{dc}$ is increased more than 50 µ A above I$_{t}$ , the op tically observed mechanical amplitude x stops increasing and is saturated by collisions of the resonator mass against elec- trode T3. This amplitude saturation is also observe d in the piezoresistance variations r$_{ac}$ . 

The observed oscillation of the heat engine is at- tributed to the thermodynamic feedback mecha nism which is schematically shown in Fig. 1d. When an al- ternating mechanical displacement x = x$_{0}$e iωt w it h fre- quency ω is applied to a piezoresistive beam with s pring 

constant k , this will not just result in an elastic force, but will also result in a resistance variation r$_{ac}$ via the piezoresistive effect. If a constant DC current I$_{dc}$ is flow- ing through the spring, the resistive heating power there- fore changes by p$_{pr}$ = I 2 $_{dc}$r$_{ac}$ and causes a temperature change, which in turn will generate a thermal expansion force F$_{te}$ that adds to the external forces F$_{ext}$ on the spring, such that kx = F$_{te}$ + F$_{ext}$ . 

The amplitude and the thermal phase-delay of the thermal expansion feedback force F$_{te}$ can be described by a complex coefficient β which is defined by F$_{te}$ = βI 2 $_{dc}$kx . The coefficient β can be calculated from the material con- stants and geometry of the spring (see Supplementary Discussion A). The response of the piezoresistive spring to an external force can now be described by a complex spring constant k ∗ eff ≡ F$_{ext}$/x = k (1 − βI 2 $_{dc}$). When the piezoresistive spring is coupled to a mass m , it forms a resonator that can store energy U$_{i}$ similar to the flywheel of a conventional heat engine. In the absence of current ( I$_{dc}$ = 0 A), the resonator’s intrinsic damping can be ex- pressed by a Q -factor Q$_{int}$ and its resonance frequency 

