6 a Engine beam T$_{b}$ Heat sink T$_{0}$ Internal energy beam C T U = ib vb b Internal energy x U = i ½k T B eff -1 0 1 Q$_{int}$ I$_{dc}$ 2 Im GLYPH<c98> 1 T$_{eff}$/T$_{b}$ Refrigerator Heat pump Heat engine b T /T =1/(1-Q$^{I}$ Im ) 2 GLYPH<c98> ef f b i nt dc Thermal conduction Q =k (T -T ) t h b 0 Fluctuation-dissipation Q$_{fd}$=½k (T -T ) /Q B eff b int GLYPH<c119>$_{0}$ W$_{e}$ External power U = ib C T vb b U = i ½k T B eff Mechanical resonance mode x T$_{eff}$ B GLYPH<c186> k<x >/k 2 Heat engine power W ½k T I he B eff dc GLYPH<c61> GLYPH<c119> GLYPH<c98> 0 Im 2 Resistive heating power Q =R I r dc dc 2 W <0 $_{e}$+U$_{i}$ W >0 $_{e}$+U$_{i}$ W >0 $_{e}$+U$_{i}$ W =Q and U + he f d W =0 e i For Q$_{int}$I 2 $_{dc}$Im β > 1, the steady state rate equation in Fig. 5a can only be satisfied with ˙ W$_{e}$ + ˙ U$_{i}$ > 0. There- fore external work W$_{e}$ is generated and the engine beam operates as a heat engine. The distinction between the work and heat flows in Fig. 5 is subtle. For T$_{eff}$ → ∞ it follows that Q$_{eff}$ tends to infinity such that the motion of the resonator is perfectly sinusoidal, like in Fig. 2c. This energy can theoretically be stored with 100% efficiency and should be called work. At any finite value of T$_{eff}$ the resonator performs a thermally driven random motion, which cannot be fully converted into work according to Carnot’s theorem 19 and should be called heat. For the miniaturization of engines a high power den- sity p$_{dens}$ is essential, since it determines the minimum size the engine can have in order to perform its task within the specified time. The maximum power ˙ W$_{he}$ that the demonstrated heat engine can generate is given by 1 $_{2}$k < x 2 > ω$_{0}$I 2 $_{dc}$Im β . In order to achieve the maximum power density, a current I$_{dc}$ =5 mA is applied for which the amplitude of the centre of mass is limited by colli- sions with electrode T3 to x =100 nm. For an optimum value of Im β =100 A − 2 (see Fig. 4c) it is found that ˙ W$_{he}$ =12.7 nW, about 4% of which is dissipated by the intrinsic losses ( ˙ Q$_{fd}$ ). It thus follows that the piezoresis- tive heat engine, with an engine displacement volume of Performance FIG. 5: Diagrams showing the heat flow and operation modes of the engine beam. a Rate equations governing the energy transport between heat sink, engine beam and resonance mode. The resistive heating ˙ Q r powers the engine beam, which converts the heat into work ˙ W$_{he}$ on the resonance mode. The resonance mode dissipates the power ˙ W$_{he}$ at a rate ˙ Q$_{fd}$ and uses the excess power to perform external work ˙ W e or to increase T eff . Heat from the engine beam flows away to the heat sink by its thermal conductance k$_{h}$ . Internal energy ( U$_{ib}$ , U i ) is stored by the heat capacitances of the beam at constant volume ( C$_{vb}$ ) and of the resonance mode ( 1 $_{2}$k B ). All quantities in this diagram are time averaged values and the dots represent time derivatives. Integration of equation (1) yields 11,14,17 the relation T eff /T$_{b}$ = Q eff /Q int and combining this with equation (2) and the definition of Q int results in the equations for ˙ Q$_{fd}$ and ˙ W$_{he}$ . b Three different regimes of operation are distinguished. Depending on the feedback factor Q int I 2 $_{dc}$Im β , the device operates as a refrigerator, heat pump or heat engine. If no external work or heat is transferred ( ˙ W e + ˙ U i =0), the effective temperature of the resonance mode follows the blue line T eff /T$_{b}$ = (1 − Q int I 2 $_{dc}$Im β ) $_{−}$1 = Q eff /Q int .