# Transimpedance Amplifier for Noise Measurements in Low-Resistance IR Photodetectors
## 2 Proposed Approach
Minimizing the ratio between the DUT thermal noise \(S_{D}\) and the background noise means minimizing the quantity \(Q_{n}\):  
\[Q_{n}=\frac{S_{iBN}}{S_{D}}=\frac{R_{D}S_{in}}{4 kT}+\frac{R_{D}S_{en}}{4 kT \left|R_{D}\right\|R_{R}\right|^{2}}+\frac{R_{D}S_{eB}}{4 kT\left|R_{D}\right\| R_{R}\right|^{2}}+\frac{R_{D}}{R_{R}} \tag{4}\]  
From Equation (4), we conclude that \(R_{D}/R_{R}<<1\) is a necessary condition to obtain \(S_{iBN}/S_{D}<<1\). If this condition is satisfied, we also have \(R_{D}\|R_{R}\approx R_{D}\) and Equation (4) can be rewritten as:  
\[Q_{n}=\frac{R_{D}^{2}S_{in}+S_{en}+S_{eB}}{4 kTR_{D}} \tag{5}\]  
If we resort to either batteries or ultra-low-noise voltage sources for obtaining the required bias for the DUT, \(S_{eB}\) can be made negligible [22]. From Equation (5), if \(S_{eB}\approx 0\), the lowest value of \(Q_{n}\), for given values of \(S_{in}\) and \(S_{en}\), is obtained when \(R_{D}=R_{Qmin}\)  
\[R_{Qmin}=\sqrt{\frac{S_{en}}{S_{in}}} \tag{6}\]Note, however, that \(R_{Qmin}\) is frequency dependent, and that the value of \(R_{D}\) is set by the DUT and cannot be easily modified. To further simplify the discussion, we will assume a \"typical\" value of \(R_{D}\) = 1 k\(\Omega\) with a thermal voltage noise of 4\(kTR_{D}\)\(\approx\) 16.6 \(\times\) 10\({}^{-18}\) V\({}^{2}\)/Hz (\(\approx\) 4 nV / \(\surd\)Hz) at room temperature. Let us first explore the possibility of using a monolithic low-noise operational amplifier with BJT (Bipolar Junction Transistor) or FET input stage technology. With BJT input stages, lower levels of input voltage noise can be achieved. On the other hand, the current noise obtained when using BJTs is typically six orders of magnitude higher than that obtained using MOSFET input amplifiers. As an example, the voltage noise of the MOSFET input TLC2201 operational amplifier at 1 Hz is 3.6 \(\times\) 10\({}^{-15}\) V\({}^{2}\)/Hz (60 nV/ \(\surd\)Hz), with a specified current noise of less than 10\({}^{-30}\) A\({}^{2}\)/Hz (1 fA/ \(\surd\)Hz). With our assumed reference \(R_{D}\), the contribution of \(S_{in}\) to \(Q_{n}\) is completely negligible (less than 10\({}^{-8}\)), but due to the value of \(S_{in}\), \(Q_{n}\) is about 225. For the low-noise BJT input OP27, the voltage noise is 100 times lower (36 \(\times\) 10\({}^{-18}\) V\({}^{2}\)/Hz) than that of the TLC2201, resulting in a contribution to \(Q_{n}\) of about 2.25. On the other hand, the effect of the current noise is not negligible as before: at the same frequency (1 Hz), its value is about 22.4 \(\times\) 10\({}^{-24}\) A\({}^{2}\)/Hz (\(\approx\) 5 pA/ \(\surd\)Hz), and this results in a further contribution of about 1.6 to \(Q_{n}\), for a total value of \(Q_{n}\) close to 4. Note, moreover, that the reference value for our chosen resistance is close to the value that minimizes \(Q_{n}\) at 1 Hz for the OP27 (\(R_{Qmin}\) = 1270 \(\Omega\)). This means that for impedance values significantly below or above 1 k\(\Omega\), \(Q_{n}\) becomes significantly greater than 4. The results discussed above are quite typical and do not change significantly across the range of commercially available monolithic low-noise operational amplifiers. To obtain a value of \(Q_{n}\) that is significantly below 1, we must therefore resort to a custom design for the operational amplifier OA1. In particular, we can take advantage of the noise characteristics of discrete component devices (BJTs or JFETs), allowing us to obtain significantly lower levels of equivalent voltage noise. The reduction in voltage noise, however, is accompanied by an increase in current noise. This means that unless we are dealing with very low impedances (well below 100 \(\Omega\)), there is no advantage in using discrete BJTs as front-end devices. On the other hand, discrete low-noise JFET devices can make it possible to reach sufficiently low equivalent input voltage noise, with the contribution of the current noise to \(Q_{n}\) remaining negligible.