## 4 Maximum Possible Controlled Cooling Rate in the Membrane-Gas System
Note that at \(R=3\cdot 10^{8}\) K/s, ballistic (uncontrolled) cooling begins at some critical temperature difference \((T_{\text{C}}-T_{t})\) of about 400 K. This critical temperature difference \((T_{\text{C}}-T_{t})\) increases with increasing \(R\), so that controlled cooling above \(5\cdot 10^{8}\) K/s is practically impossible, even with a scanning interval as large as \(\Delta T=1000\) K.  
However, the controlled cooling rate can be increased by additional heat transfer in a membrane with a higher thermal conductivity \(\lambda_{m}\). The thermal conductivity \(\lambda_{m}\) can be increased by depositing an additional heat-conducting Au layer on the membrane. This idea was successfully demonstrated on a commercially available chip calorimeter with a hot zone radius \(r_{0}\) of about 5 um [23]. A rate of about \(10^{8}\) K/s was achieved in helium gas for a sensor with an additional heat-conducting Au layer on the membrane [23]. In this article, we estimate the maximum possible cooling rate as \(r_{0}\) decreases. In fact, for an additional Au layer with a thickness \(d_{Au}\) and thermal conductivity \(\lambda_{Au}=317\) W/m\(\cdot\)K [33], we obtain a membrane with \(\lambda_{m}=\lambda_{Au}(d_{Au}/d_{0})+\lambda_{0}(d/d_{0})\) in the lateral direction of the membrane, where \(d_{0}\) is the thickness of the silicon nitride layer. Suppose \(d_{0}=0.8\) um and \(d_{Au}=0.4\) um,  
Figure 3: Normalized scanning rate \(\partial T/\partial t\) vs. temperature at the center of the membrane for the power \(P(t)\) shown in the inset (**a**) and the temperature profile \(T(t_{0},r)\) (**b**) for \(R=10^{8}\) K/s and \(3\cdot 10^{8}\) K/s—black squares and red circles, respectively. The heater acts uniformly in the ring \(r\in(r_{1},r_{0})\) at \(r_{1}=2r_{0}/3\). Thermal conductivity of the membrane \(\lambda_{m}=3\) W/m\(\cdot\)K.  
then \(d_{m}=1.2\)\(\upmu\)m, \(\lambda_{m}=100\) W/m\(\cdot\)K, and the average volumetric heat capacity of the membrane is \(\rho_{m}c_{m}=2.4\cdot 10^{6}\) J/m\({}^{3}\)K at \(\rho_{Au}c_{Au}=2.5\cdot 10^{6}\) J/m\({}^{3}\)K for Au [33]. In this case, controlled cooling is possible at \(R=10^{9}\) K/s and even at \(6\cdot 10^{9}\) K/s, see Table 3.0 The calculations were carried out for helium gas with \(\lambda=0.156\) W/m\(\cdot\)K at \(r_{0}=1\)\(\upmu\)m, \(r_{1}=2r_{0}/3\), \(d_{m}=1.2\)\(\upmu\)m, \(\lambda_{m}=100\) W/m\(\cdot\)K, \(D_{m}=4.2\cdot 10^{-5}\) m\({}^{2}\)/s, \(\rho_{m}c_{m}=2.4\cdot 10^{6}\) J/m\({}^{3}\)K, and \(\Delta T=1000\) K. The temperature profile \(T(t_{0},r)\) was calculated at the end of heating for \(t_{0}=1.03\)\(\upmu\)s and \(0.177\)\(\upmu\)s at \(R=10^{9}\) K/s and \(6\cdot 10^{9}\) K/s, respectively. The maximum possible rate of controlled cooling can be even higher if the heater is uniformly distributed over the entire hot zone for \(r\in(0,r_{0})\). However, in this case, the temperature profile in the middle of the hot zone is not as flat as in the case of the ring heater, see Figures 4 and 5. Controlled cooling is possible at \(R=10^{9}\) K/s, \(6\cdot 10^{9}\) K/s, and even \(10^{10}\) K/s, see Table 3.1 The calculations were carried out for helium gas with \(\lambda=0.156\) W/m\(\cdot\)K at \(r_{0}=1\)\(\upmu\)m, \(d_{m}=1.2\)\(\upmu\)m, \(\lambda_{m}=100\) W/m\(\cdot\)K, \(D_{m}=4.2\cdot 10^{-5}\) m\({}^{2}\)/s, \(\rho_{m}c_{m}=2.4\cdot 10^{6}\) J/m\({}^{3}\)K, and \(\Delta T=1000\) K. The temperature profile \(T(t_{0},r)\) was calculated at the end of heating for \(t_{0}=1.03\)\(\upmu\)s, \(0.175\)\(\upmu\)s, and \(0.105\)\(\upmu\)s at \(R=10^{9}\) K/s, \(6\cdot 10^{9}\) K/s, and \(10^{10}\) K/s, respectively. Table 3.2 Normalized scanning rate \(\partial T/\partial t\) vs. temperature in the center of the membrane for the power \(P(t)\) shown in the inset (**a**) and the temperature profile \(T(t_{0},r)\) (**b**) for \(R=10^{9}\) K/s and \(6\cdot 10^{9}\) K/s—black squares and red circles, respectively. The heater acts uniformly in the ring \(r\in(r_{1},r_{0})\) at \(r_{1}=2r_{0}/3\).