4 1.2540 1.2560 f (MHz) 3.0 mA 2.4 1.6 mA 1.0 0.5 mA 0.2 1.2550 1.2552 1.2554 1.2556 f (MHz) 10 -16 10 -14 10 -12 10 -10 2< v GLYPH<c119> 2 > (V 2 /Hz) 1.045 mA 0.8 mA 0.6 0.4 mA 0.2 a b c Z = load GLYPH<c165> Z = load 40 H GLYPH<c109> Z = load GLYPH<c165> Z = load 40 H GLYPH<c109> 1/70 1/80 1/90 1/100 1/300 1/500 1/10 3 1/T (K ) eff -1 0 2 4 6 8 I (mA ) dc 2 FIG. 3: Voltage spectral density <v 2 $_{ω}$> for different values of I$_{dc}$ . Measurements by a spectrum analyser using the circuit in Fig. 1c show that a in the open state ( Z load = ∞ ) the thermal motion is amplified when the heat engine is driven by a current I$_{dc}$ . b For Z load = 40 µ H, the mechanical energy decreases with increasing current and the device operates as a refrigerator. Note the factor 4 difference in frequency scale. c 1 /T eff as determined from the data in 3a and 3b by evaluating 1 $_{2}$k B T eff = 1 $_{2}$k <x$^{2}$> , where <x $^{2}$> is determined by integration of the fitted dashed lines (see Methods). The difference between the y -axis intersection of the data and the ambient temperature is within experimental uncertainty limits. displacement noise density <x 2 $_{ω}$> of this motion near the resonance frequency in the presence of the feedback force F$_{te}$ can be determined from the fluctuation-dissipation theorem $^{11,14,17}$: temperature T$_{eff}$ of the resonance mode either increases like in a heat pump (Fig. 3a) or is refrigerated down to an effective temperature of 70 K at I$_{dc}$ =3.0 mA (Fig. 3b). The difference between the spectra in Figs. 3a and 3b is caused by the fact that the feedback coefficient β is proportional to the ratio γ$_{Z}$ ≡ p$_{pr}$/ ( I 2 $_{dc}$r$_{ac}$ ) between the piezoresistive heating power p$_{pr}$ and the AC resis- tance r$_{ac}$ as can be seen from Fig. 1d. Therefore the coefficient β can be written as the product of two com- plex numbers: β = γ$_{Z}$β$_{0}$ , where β$_{0}$ depends solely on the device geometry and material parameters and γ$_{Z}$ can be controlled externally via Z$_{load}$ in the following way. Figure 1c shows that a resistance change r$_{ac}$ of the beam will induce both an AC current i$_{ac}$ = − v$_{ac}$/Z$_{tot}$ and AC voltage v$_{ac}$ = R$_{dc}$i$_{ac}$ + I$_{dc}$r$_{ac}$ . The total heat- ing power P$_{dc}$ + p$_{pr}$ in the engine beam is given by ( I$_{dc}$ + i$_{ac}$ ) $^{2}$( R$_{beam}$ + r$_{ac}$ ) and for r$_{ac}$ ≪ R$_{beam}$ it follows that γ$_{Z}$ = 1 − 2 R$_{beam}$/ ( Z$_{tot}$ + R$_{dc}$ ). Where k$_{B}$ is Boltzmann’s constant and it is assumed that Q$_{eff}$ ≫ 1 and | βI 2 $_{dc}$|  ≪ 1. To determine <x 2 $_{ω}$> , the AC voltage spectrum <v 2 $_{ω}$> , which is proportional to <x 2 $_{ω}$> , is measured (Fig. 3a,b) at several values of I$_{dc}$ by a spectrum analyser using the measurement circuit in Fig. 1c. In Fig. 3b an in- ductor Z$_{load}$ = 40 µ H is placed in parallel to the struc- ture. Equation (1) is used to fit the data (see Meth- ods). The measurements in Fig. 3a show that even be- low the threshold current I$_{t}$ the thermal fluctuations in the resonance mode are amplified. It is observed that the stored thermal energy in the resonance mode in- creases with DC current, such that it equals the energy which would be stored in the mode if the resonator would be at an effective temperature T$_{eff}$ which is defined by 1 $_{2}$k$_{B}$ T$_{eff}$ ≡ 1 $_{2}$k ∫ ∞ −∞ <x 2 $_{ω}$> d ω 2 π = 1 $_{2}$k <x$^{2}$> . Similar to res- onators cooled by optical or RF methods 11–18 the tem- perature T$_{eff}$ represents the mechanical energy stored in the resonance mode corresponding to the degree of free- dom x and is not representative for the other degrees of freedom of the resonator. By integrating the fitted <x 2 $_{ω}$> curves, T$_{eff}$ is determined and is plotted in Fig. 3c. Depending on the load impedance Z$_{load}$ , the effective Voltage spectra like those in Fig. 3a,b are recorded for several real and imaginary values of Z$_{load}$ , using resistors, capacitors and inductors. The fitted values of Q$_{eff}$ are plotted in Fig. 4a and b. For small values of Z$_{load}$ a more accurate determination of Q$_{eff}$ was made by generating a random electrostatic noise force on the resonator using a white voltage noise source connected to terminal T3. The observed linear dependence of 1 /Q$_{eff}$ on I 2 dc in Fig. 4a,b corresponds well with equation (2). From the slope Quantitative analysis <x 2 $_{ω}$> = 2 ω 3 $_{0}$k$_{B}$ T$_{b}$/ ( kQ$_{int}$ ) ( ω 2 − ω 2 $_{0}$) 2 + ( ωω$_{0}$/Q$_{eff}$ ) 2 (1) 1 Q$_{eff}$ = 1 Q$_{int}$ − I 2 $_{dc}$Im β (2)