7 Methods the engine beam of V = 0 . 34 µ m $^{3}$, has a power density p$_{beam}$ = ˙ W$_{e}$/V =37 GW/m 3 which is almost a factor 1000 higher than that of car engines which have typical power densities of around 50 MW/m $^{3}$. The power density p$_{beam}$ of the piezoresistive heat engine is also high when com- pared to other microengines $^{1,20,21}$, which have reported power densities up to 3 GW/m $^{3}$. However, in some of these cases the power density is defined using the vol- ume of the total structure instead of the engine displace- ment volume. Due to the relatively large size of the proof mass compared to the engine beam size, this definition of power density is less favourable for the piezoresistive heat engine, yielding p$_{tot}$ =11 MW/m $^{3}$. Using the equations from Fig. 5a, the efficiency of the presented engine is calculated to be η$_{he}$ = ( ˙ W$_{he}$ − ˙ Q$_{fd}$ ) / ˙ Q$_{r}$ =6.4 × 10 − $^{7}$. Besides being limited by the Carnot theorem, the efficiency of the piezoresistive heat engine is also limited by the fracture limit of silicon to around η$_{pr}$$_{,}$$_{Si}$$_{,}$$_{max}$ = 3 . 4 × 10 − $^{4}$, as analysed in the Supple- mentary Discussion B. It is therefore expected that the efficiency and power density of the piezoresistive heat en- gine can still be improved by a factor ∼ 500 by increasing the displacement amplitude and thermal design of the engine. Even after optimization, this efficiency is low compared to that of micro- and nano-engines driven by electrostatic forces $^{22–25}$. Nevertheless, the comparison in Supplementary Discussion B shows that piezoresistive heat engines have the potential to outperform electro- static engines in terms of power density. This high power density and the absence of transistors can facilitate the development of mechanical actuator 26 and sensor 10,27 ar- rays with very high element densities. Structure The piezoresistive heat engine in Fig. 1a is made out of a 1.5 µ m thick silicon layer on a silicon- on-insulator wafer, with a phosphor doping concentra- tion N$_{d}$ =4.5 × 10 18 cm − 3 giving a specific resistivity of ρ$_{dc}$ =10 − $^{4}$Ωm. This doping concentration results in a low temperature coefficient of resistivity 28 of less than 0.1% per K (see Supplementary Fig. C). The thin crys- talline silicon layer is structured in a single mask step by a deep reactive ion etch and the buried SiO$_{2}$ layer below the mass and beams is removed in a hydrogen flu- oride vapour etch. A DC current I$_{dc}$ is driven through the beams via terminals T1 and T2 as shown in Fig. 1b and c. As a result of the geometry of the structure, the current density, heating power density and mechanical stress are concentrated in the narrow engine beam, but the resonance frequency of the resonator is mainly de- termined by the spring beam and resonator mass. The in-plane mechanical bending resonance mode that de- termines the fundamental resonance frequency f$_{0}$ =1.26 MHz has a spring constant k =256 N/m as determined from finite element method (FEM) simulations. Mea- surements are performed at 315 K in a vacuum chamber at a pressure below 10 − 2 mbar. External excitation The in-plane fundamental me- chanical resonance of the structure at 1.26 MHz is char- acterized by actuating it via terminal T3 with an AC electrostatic force F$_{ac}$$_{,}$$_{act}$ = ε$_{0}$AV$_{dc}$$_{,}$$_{act}$v$_{ac}$$_{,}$$_{act}$/g 2 gener- ated by a voltage V$_{dc}$$_{,}$$_{act}$ + v$_{ac}$$_{,}$$_{act}$ on terminal T3, with V$_{dc}$$_{,}$$_{act}$ = -1 V. The actuation gap g =200 nm has an ac- tuation area A = 60 × 1 . 5 µ m $^{2}$. A DC current I$_{dc}$ = 0 . 1 mA is applied as in Fig. 1b. The displacement of the centre of mass x is proportional to the measured piezore- sistive AC current i$_{pr}$ , which is shown in Fig. 2a. The reduction of the intrinsic Q -factor Q$_{int}$ upon increasing the pressure in the vacuum chamber confirms the me- chanical nature of the resonance. The solid fits in Fig. 2a allow us to determine the piezoresistive factor K$_{pr}$ , which is defined by r$_{ac}$/R$_{dc}$ = K$_{pr}$x , where r$_{ac}$ is the amplitude of the piezoresistance variation and R$_{dc}$ is the total DC resistance. From the maximum piezoresistive current i$_{pr,peak}$ the piezoresistive factor is determined to be K$_{pr}$ = − i$_{pr,peak}$ I$_{dc}$ k F$_{ac}$$_{,}$$_{act}$Q$_{int}$ = − 6 . 6 × 10 5 m − $^{1}$. Spectrum analysis The spectra in Fig. 3 and 4 are measured using the electrical circuit in Fig. 1c. Be- cause the displacement of the centre-of-mass x$_{ω}$ ≡ x ( ω ) causes a piezoresistive voltage v$_{ω}$ ≡ v$_{ac}$ = [ $_{Z}$− 1 tot + $_{R}$− 1 dc ] − $^{1}$I$_{dc}$ K$_{pr}$x (Fig. 1c), the electrical spectrum <v 2 $_{ω}$> measured by the spectrum analyser is proportional to equation (1). The voltage spectral density in Fig. 3a and b is fit by dashed curves which are given by equation (1) with <v 2 $_{ω}$> = I 2 $_{dc}$K 2 $_{pr}$<x 2 $_{ω}$>/ [ $_{Z}$− 1 tot + $_{R}$− 1 dc ] 2 and which include a fit parameter to account for the white John- son noise background. The total AC impedance parallel to the resonator Z$_{tot}$ = ( iωC$_{0}$ ) − 1 + ( iωC$_{1}$ + $_{Z}$− 1 $_{load}$) − 1 can be controlled by adjusting Z$_{load}$ . The DC resis- Similar to other recently investigated cooling methods $^{11–18}$, the coefficient of performance of the refrigerator η$_{cool}$ = − ˙ W$_{he}$/ ˙ Q$_{r}$ is very small when T$_{eff}$ is at or below room temperature. The refrigerator will therefore not be useful for lowering the temperature of systems with many degrees-of-freedom. However, because the heat capacity ∂U$_{i}$ ∂T$_{eff}$ of a single-degree-of- freedom resonance mode is only 1 $_{2}$k$_{B}$ and because the thermal insulation of the mode is high as a result of its high Q$_{int}$ (see Fig. 5a), the piezoresistive refrigerator can still reduce T$_{eff}$ by more than 200 K. The Brownian fluctuations associated with the energy 1 $_{2}$k$_{B}$ T$_{eff}$ often dominate the noise in mechanical sensors and mirrors and their reduction can thus increase the signal-to-noise ratio and might ultimately enable quantum-limited measurement and control of mechanical resonators. The presented piezoresistive refrigerator provides an alternative cooling method for reducing these Brownian fluctuations, which has the advantage that it does not require laser or RF sources and only requires a silicon chip with a relatively low DC power.