# Instability and Drift Phenomena in Switching RF-MEMS Microsystems
## 4 Instabilities Due to Temperature Variations  
Variations of temperature are known to cause instability phenomena in suspended beams. The most important source of these instabilities is the difference in the linear thermal expansion (LTE) coefficient of the materials used for the suspended structure and the substrate. Switching beams are often made of metal, typically gold or aluminum, while the substrate is generally silicon or, for high-frequency switching, quartz. Both substrates have an LTE coefficient much lower than that of gold or aluminum. When the temperature rises, the beam expands more than the substrate. If the beam is a cantilever, this does not cause relevant effects on the geometry and the internal stress of the structure, because the beam is free to expand. Only minor changes in the switch parameters are expected in this case, due to the change in elastic modulus of the materials or stress gradient variations [28]. In the case of a double-clamped beam, the beam expansion is constrained by the supports and the beam experiences a compressive thermal stress which adds to the residual stress of the beam according to the following equation  
\[\sigma_{total}=\sigma_{Tamb}-E(\alpha_{beam}-\alpha_{substrate})\Delta T \tag{3}\]  
where \(\sigma_{total}\) and \(\sigma_{Tamb}\) are the beam's total stress and residual stress at room temperature, \(E\) is the beam's elastic modulus, \(\Delta T\) is the temperature difference, and \(\alpha_{beam}\) and \(\alpha_{substrate}\) are the LTE coefficients of the suspended beam and the substrate, respectively. The variations of beam shape, internal stress, and actuation voltage as a function of temperature are schematized in Figure 4.  
When the temperature is high enough, the beam can no longer sustain the compression and buckles, assuming the typical out-of-plane curved shape. Buckling occurs when the temperature is higher than a threshold value, called the critical temperature [29]. The critical temperature depends on the geometry of the beam and on the room temperature residual stress of the beam, which is normally tensile. The more tensile the beam stress at room temperature, the higher the buckling temperature. However, the room temperature tensile stress is determined by the fabrication process conditions and is quite difficult to modify or control. Moreover, the higher the tensile stress, the higher the actuation voltage of the beam, since the stress influences its spring constant [25]. If we look at the actuation voltage variation with the temperature, we see that when the temperature increases, in the first phase the actuation and release voltages decrease, due to the reduction of the internal tensile stress. The actuation voltage reduction is roughly dependent on the square root of the temperature difference when the residual stress is high, as can be seen from Equation (4). Figure 4: Scheme of the interrelation between temperature, beam shape, internal stress, and actuation voltage in a clamped–clamped beam. The precise value of the critical temperature depends on the beam’s geometry and material. \[V_{act}\sim\sqrt{k_{elastic}(E)+B[\sigma_{Tamb}-E(\alpha_{beam}-\alpha_{substrate}) \Delta T]} \tag{4}\]  
where \(k_{elastic}(E)\) is the part of the spring constant which is stress independent and determined only by the elastic modulus \(E\) and \(B\) is a constant which depends on the beam geometry. A detailed analytical derivation of this formula is reported in [30]. When the internal stress is approximatively zero or slightly positive, the minimum actuation voltage is reached. After that, we enter the buckling zone, and the actuation becomes unstable and difficult to calculate. The reason for this instability is mainly mechanical. In order to minimize the internal compressive stress, the suspended membrane deforms in the out-of-plane direction and changes the value of the air gap, which is no longer a constant in Equations (1) and (2).