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## Supplementary Information Piezoresistive heat engine and refrigerator 

In this Supplementary Information the thermodynamic coupling between stress and strain in piezoresistive ma- terials carrying a current is discussed in more detail. In Supplementary Discussion A an analytical expression is derived for the effective complex Young’s modulus Y ∗ $_{eff}$. This expression is used to provide an analytical estimate for the feedback coefficient β , which expresses the effect of a DC current on the effective spring constant k ∗ eff of a piezoresistive spring. The analytical result is compared to measurements presented in the main manuscript. In Supplementary Discussion B the optimal efficiency and power density of silicon heat engines is estimated. In Supplementary Figure C a measurement of the resistance of the device as a function of DC current and chuck temperature is used to estimate the temperature rise in the engine beam. 

## SUPPLEMENTARY DISCUSSION A: Quantitative discussion of the feedback mechanism and coefficient β 

To evaluate the effect of a DC electric current on the mechanical properties of a piezoresistive solid, consider a solid through which a current density J$_{dc}$ is flowing in the y -direction. When a small uniaxial vibrational stress σ$_{ac}$e iωt with frequency ω is present along the y -direction the piezoresistive effect will induce an AC resistivity change ρ$_{ac}$ : 

<img src='content_image/394309.jpg'>

Where ρ$_{dc}$ is the unstressed resistivity and π$_{l}$ is the longitudinal piezoresistive coefficient $^{1}$. It is assumed that ρ$_{ac}$ ≪ ρ$_{dc}$ . This resistivity change modifies the AC resistive heating power density p$_{pr}$ : 

<img src='content_image/394308.jpg'>

As discussed in the main manuscript, γ$_{Z}$ is a correction factor which is needed if a finite impedance to ground is present parallel to the piezoresistive resonator, such that AC currents i$_{ac}$ can also contribute to p$_{pr}$ . From the heat equation p = − k$_{h}$ ∇ $^{2}$T + c$_{p}$ρ$_{d}$∂T /∂t it follows that the heating power causes sinusoidal temperature fluctuations with amplitude T$_{ac}$ : 

<img src='content_image/394307.jpg'>

The specific heat capacity is given by c$_{p}$ and ρ$_{d}$ is the mass density. The effect of thermal heat conductivity k$_{h}$ is captured by the factor γ$_{h}$ : 

<img src='content_image/394306.jpg'>

The factor γ$_{h}$ depends on the resonator’s geometry as will be discussed below. The temperature increase generates a thermal expansion stress: 

<img src='content_image/394305.jpg'>

Y is Young’s modulus and α$_{te}$ is the thermal expansion coefficient. This thermal expansion stress adds to the externally applied stress σ$_{ac,ext}$ , such that the total stress is given by σ$_{ac}$ = σ$_{ac,ext}$ + σ$_{ac,te}$ , and the corresponding strain is given by ε$_{ac}$ = σ$_{ac}$/Y . The stress-strain relation of the piezoresistive solid is therefore the same as that of a solid with an effective complex Young’s modulus Y ∗ eff = σ$_{ac,ext}$/ε$_{ac}$ . The variables, coupling mechanisms and multiplicative factors in equations (A1-A5) have been schematically shown in figure A1 and using these equations the effective modulus can be expressed as: 

<img src='content_image/394304.jpg'>

This relation is only valid for small stress and strain, such that the linear approximation is valid and material parameters are constant. When all constants in equation (A6) are real and positive, equation (A6) is identical to the Young’s modulus of the standard anelastic solid $^{2}$, however if some of the constants are negative or have a non-zero imaginary part, it can lead to properties like mechanical self-amplification and negative creep. This modification of the effective Young’s modulus can occur in any piezoresistive solid in the presence of electrical current. Besides its effect on the dynamics of mechanical structures it might thus also affect the propagation of acoustic waves in solids. 

