content_image/400182.jpg
of these linear fits, Im β is determined and is plotted against γ$_{Z}$ as symbols in Fig. 4c. As shown by solid and dashed lines in Fig. 4c, an excellent multiple linear regression fit of the data is obtained using the function Im β = Im ( γ$_{Z}$β$_{0}$ ), which yields the fit parameters β$_{0}$ = − 123 . 7 + 64 . 6 i A − 2 and R$_{beam}$ = 439 . 3 Ω. 

In Supplementary Discussion A an analytical model is derived for the complex Young’s modulus and spring constant k ∗ eff and it is used to derive an estimate for β$_{0}$ , which yields β$_{0}$ = − 132 + 157 i A − $^{2}$. A finite element simulation of the full geometry including the anisotropy of the silicon crystal results in β$_{0}$ = − 111+58 i A − $^{2}$. Both the excellent fits in Fig. 3 and 4, and the quantitative agreement between the measured and simulated values of β$_{0}$ support the proposed feedback mechanism in Fig. 1d. 

<img src='content_image/400182.jpg'>

In order to gain insight in the operation of the heat en- gine, a schematic showing the rate equations for the flow of heat and work between electrical source, heat sink, en- gine beam and resonance mode is shown in Fig. 5a. All rates shown in this figure are time averaged, such that all variations which cancel each other during a single cycle are averaged out. The three modes of operation which were experimentally observed in Fig. 2d, 3a, and 3b are shown in Fig. 5b. For Im β < 0, ˙ W$_{he}$ is negative, thus cooling the effective temperature of the resonance mode and operating as a refrigerator. For 0 < Q$_{int}$I 2 $_{dc}$Im β < 1 heat is pumped into the resonance mode, increasing T$_{eff}$ until a new steady state is reached at which ˙ W$_{e}$ + ˙ U$_{i}$ = 0. In this mode the engine beam operates as a heat pump. 

## Flow rates of heat and work 

