# A Distributed Terahertz Metasurface with Cold-Electron Bolometers for Cosmology Missions
## 3 Noise Analysis  
In our noise analysis of the 2D CEB array we use the basic CEB model with a strong electrothermal feedback due to electron cooling [15; 16] and the concept of series arrays in a current-biased mode [9]. The operation of a CEB array can be analyzed using the heat balance equation for a single CEB, taking into account the power distribution between the \(N\times W\) bolometers:  
\[P_{cool}(V,T_{e},T_{ph})+\Sigma\Lambda(T_{e}^{5}-T_{ph}^{5})+\frac{V^{2}}{R_{j }}+I^{2}R_{a}=\left[P_{0}+\delta P(t)\right]/W/N \tag{2}\]  
Here, \(\Sigma\Lambda(T_{e}^{5}-T_{ph}^{5})\) is the heat flow from the electron to the phonon subsystems in the absorber, \(\Sigma\) is a material constant, \(\Lambda\) is the volume of the absorber, \(T_{e}\) and \(T_{ph}\) are, respectively, the electron and phonon temperatures of the absorber, \(P_{cool}(V,T_{e},T_{ph})\) is the cooling power of the \(SIN\) tunnel junction, \(R_{j}\) is the sub-gap resistance of the tunnel junction, \(R_{a}\) is the resistance of the absorber, \(\delta P(t)\) is the incoming \(RF\) power changes and \(P_{0}\) is absorbed signal power. We can separate Equation (2) into the time independent term,  
\[P_{SIN0}(V,T_{e0},T_{ph})+\Sigma\Lambda(T_{e0}^{5}-T_{ph}^{5})=P_{0}/W/N \tag{3}\]  
and the time dependent term,  
\[(\partial P_{SIN}/\partial T+5\Sigma\Lambda T_{e}^{4})\delta T=\delta P_{1} \tag{4}\]  
The first term in Equation (4), \(G_{SIN}=\partial P_{SIN}/\partial T\), is the cooling thermal conductance of the SIN junction that gives the negative electrothermal feedback (ETF); when this term is large, it reduces the temperature response, \(\delta T\), because the cooling power, \(P_{SIN}\), compensates for the change of signal power in the bolometer. The second term in Equation (4), \(G_{e-ph}=5\Sigma\Lambda T_{e}^{4}\), is the electron-phonon thermal conductance of the absorber. Figure 6: (**a**) _Pabs_ of mode#1 and mode#2, (**b**) _Pabs_\({}^{orb}\) of the unit cell for arbitrary polarization. From Equation (4), we can define an effective complex thermal conductance that controls the temperature response of the CEB to the incident signal power:  
\[G_{eff}=G_{SIN}+G_{e-ph} \tag{5}\]  
In analogy with TES [10], the effective thermal conductance of the CEB is increased by electron cooling (negative ETF). Here, we assume that the \(SIN\) tunnel junctions are current-biased, and the voltage is measured by a JFET amplifier. The responsivity, \(S_{V}\), is described by the voltage response to the incoming power:  
\[S_{V}=\frac{\delta V_{\omega}}{\delta T_{\omega}}=\frac{\partial V/\partial T }{G_{e-ph}+G_{SIN}} \tag{6}\]  
In the second term of Equation (7),  
\[G_{SIN}=\frac{\partial P_{SIN}}{\partial T}-\frac{\partial P_{SIN}}{\partial V }\bigg{(}\frac{\partial I}{\partial T}/\frac{\partial I}{\partial V}\bigg{)} \tag{7}\]  
is the cooling thermal conductance of the \(SIN\) junction, which provides some electron cooling and helps to avoid the overheating of the absorber. Noise properties are characterised by the noise equivalent power (\(NEP\)), which is the sum of three contributions:  
\[NEP_{tot}^{2}=N*W*NEP_{e-ph}^{2}+N*W*NEP_{SIN}^{2}+NEP_{AMP}^{2} \tag{8}\]  
here, \(NEP_{e-ph}\) is the noise associated with electron-phonon interaction:  
\[NEP_{e-ph}^{2}=10k_{B}\Sigma\Lambda(T_{e}^{6}+T_{ph}^{6}) \tag{9}\]  
In Equation (8), \(NEP_{SIN}\) is the noise of the \(SIN\) tunnel junctions. The \(SIN\) noise has three components: the shot noise, \(2 el/S2l\), the fluctuation of the heat flow through the tunnel junctions, and the correlation between these two processes:  
\[NEP_{SIN}^{2}=\frac{\delta l_{\omega}^{2}}{\big{(}\frac{\partial I}{\partial V }S_{V}\big{)}^{2}}+2\frac{\big{\langle}\delta P_{\omega}\delta I_{\omega} \big{\rangle}}{\frac{\partial I}{\partial V}S_{V}}+\partial P_{\omega}^{2} \tag{10}\]  
Due to this correlation, the shot noise is increased by 30-50% in contrast to a CEB in voltage-biased mode, where strong anti-correlation decreases the shot noise.