11 Substituting these values in equations (A13) and (A8) and using γ$_{k}$ =0.11 from equation (A9) yields β$_{0}$ = − 132 + 157 i A − $^{2}$. This value is higher than the experimental value, which is mainly attributed to the fact that the actual heat flux outside the beam is not 1-dimensional but radial, which results in a larger value of γ$_{h}$ . A finite element simulation of the full geometry, including the anisotropy of the silicon crystal results 3 in β$_{0}$ = − 111 + 58 i A − $^{2}$, in good agreement with β$_{0}$ = − 123 . 7 + 64 . 6 i A − 2 as determined from the fits of the measurements in figure 4 of the main manuscript. The optimal efficiency of any heat engine η$_{he,max}$ is limited by the Carnot efficiency η$_{C}$ = 1 − T$_{min}$/T$_{max}$ . In solid heat engines there is another efficiency constraint related to the maximum strain before fracture of the solid working substance. To illustrate this, consider a silicon beam with cross section A and length L going through an idealized Stirling cycle of heating at constant volume, expansion at constant high temperature T$_{max}$ = T$_{min}$ + Δ T , cooling at constant volume and contraction at constant low temperature T$_{min}$ . The beam’s spring constant is k = Y A/L and the static thermal expansion is Δ x$_{te}$ = α$_{te}$ Δ T L . The heat needed to raise the temperature of the beam is Q$_{h}$ = ALc$_{v}$ρ$_{d}$ Δ T , where c$_{v}$ is the heat capacity at constant volume. The isothermal expansion over a distance Δ x cost external work W$_{heat}$$_{,}$$_{e}$ = − 1 $_{2}$k (Δ x − Δ x$_{te}$ ) $^{2}$. To keep the temperature stable during expansion, additional heat ∼ Q$_{h}$ ( c$_{p}$ − c$_{v}$ ) /c$_{v}$ is needed. Then the beam is cooled at constant volume to temperature T$_{min}$ . During isothermal contraction, the beam generates external work W$_{cool}$$_{,}$$_{e}$ = 1 $_{2}$k (Δ x ) $^{2}$. The total external work done by the engine beam is therefore W$_{e}$ = W$_{heat}$$_{,}$$_{e}$ + W$_{cool}$$_{,}$$_{e}$ ≈ k Δ x Δ x$_{te}$ = α$_{te}$Y A Δ x Δ T , for Δ x$_{te}$ ≪ Δ x . The fracture strain limit of silicon which is in the range 4 of ε$_{Si}$$_{,}$$_{frac}$ ≈ 0.01-0.03, as a conservative estimate we use ε$_{Si,frac}$ = 0.01. The maximum expansion Δ x from compressive to tensile strain is given by be Δ x$_{max}$ = 2 ε$_{Si}$$_{,}$$_{frac}$L . Therefore the optimal efficiency of a silicon thermal engine is estimated to be: Higher efficiencies are possible if the strain is always kept compressive to enable higher Δ x , or by using a regenerator to reuse the heat lost during cooling. Note that for quasi-static thermal actuators Δ x = Δ x$_{te}$ and the work done is W$_{e}$ = 1 $_{2}$k (Δ x$_{te}$ ) $^{2}$, such that the optimal efficiency is lowered by a factor α$_{te}$ Δ T/ (4 ε$_{Si}$$_{,}$$_{frac}$ ). Even for Δ T =100 K, a quasi-static actuator will have an optimal efficiency that is a factor 150 lower than the optimal dynamic effiency. For most quasi-static actuators the efficiency will be much lower because the thermal conduction rate ˙ Q$_{t}$ will not only cool during the isothermal contraction, but during the whole cycle. Now we continue to estimate the maximum efficiency of the piezoresistive heat engine. From figure 5a in the main manuscript it is found that the maximum rate of work done by the heat engine is given by ˙ W$_{he}$ = 1 $_{2}$k < x 2 > ω$_{0}$I 2 $_{dc}$Im β . The value of β in equation (A8) is maximal for n-type silicon ( π$_{l}$ < 0) with γ$_{k}$ = 1, γ$_{Z}$ = 1. Optimally γ$_{h}$ is zero, however this would mean that there is no thermal conduction and no cooling of the heat engine. As a compromise γ$_{h}$ = ωρ$_{d}$c$_{p}$ is taken, reducing the efficiency by a factor 2. Thus it is found from equation (A8) that the maximum value of Im β for the presented piezoresistive heat engine is given by: The highest measured value of Im β is a factor 75 lower, of which a factor 9 can be attributed to γ$_{k}$ = 0 . 11, and the remaining factor ∼ 8 is mainly due to a too high value of γ$_{h}$ . The heating power in the beam is given by ˙ Q$_{r}$ = I 2 $_{dc}$ρ$_{dc}$ L/A . The spring constant is given by k = Y A/L and the average amplitude < x 2 > = 1 $_{2}$( ε$_{Si}$$_{,}$$_{frac}$L ) $^{2}$. Thus the optimal efficiency of the piezoresistive heat engine is found to be: A small value value of γ$_{k}$ does not reduce the efficiency, because the reduction in β is compensated by the higher value of stored energy 1 $_{2}$k < x 2 > in the equation for ˙ W$_{he}$ . In other words, the efficiency of the heat engine beam is independent of the resonator to which it is coupled. For the heat engine reported in the main manuscript, the maximum strain in the engine beam is estimated with finite element simulation to be ε$_{b}$ ≈ 0 . 0015 when operating at an amplitude of the center of mass x =100 nm. Moreover the measured Im βγ$_{k}$ is a factor 8 lower than optimal, such that the estimated efficiency is η$_{he}$$_{,}$$_{est}$ = η$_{pr}$$_{,}$$_{Si}$$_{,}$$_{max}$ × (0 . 0015 / 0 . 01) $^{2}$/ 8 = 10 − 6 close to the measured value. The difference between the measured efficiency and equation (B3) suggest that there is still room for efficiency improvement by a factor 500 of the presented heat engine concept by increasing its amplitude and reducing the heat conductivity Im β$_{pr}$$_{,}$$_{max}$ ≈ − ρ$_{dc}$π$_{l}$α$_{te}$Y 2 A $^{2}$ωρ$_{d}$ c$_{p}$ $_{= 7500 A}$− 2 (B2) η$_{pr}$$_{,}$$_{Si}$$_{,}$$_{max}$ = ˙ W$_{he}$ ˙ Q$_{r}$ ≈ − π$_{l}$Y $^{2}$α$_{te}$ ε 2 Si , frac 8 ρ$_{d}$c$_{p}$ = 3 . 4 × 10 − 4 (B3) η$_{Si}$$_{,}$$_{max}$ = W$_{e}$/Q$_{h}$ ≈ 2 Y α$_{te}$ε$_{Si}$$_{,}$$_{frac}$ ρ$_{d}$c$_{p}$ = 4 × $_{10}$− 3 (B1) SUPPLEMENTARY DISCUSSION B: Efficiency and power density of silicon heat engines