# Transimpedance Amplifier for Noise Measurements in Low-Resistance IR Photodetectors
## 2 Proposed Approach  
To better understand the challenges that must be faced when designing a transimpedance amplifier for performing current noise measurements on a low-impedance DUT, the simplified circuit presented in Figure 1 is considered. In this system, the device under test (Z\({}_{D}\)) is biased at a constant voltage \(V_{B}\) because of the virtual short circuit between the inputs of the operational amplifier \(OA_{1}\). The transimpedance gain between the DUT current source \(i_{D}\) and the output voltage of \(OA_{1}\) is set by the resistor \(R_{R}\). In choosing the value of \(R_{R}\), the linearity range of \(OA_{1}\) must be taken into account, since the DC current flowing through the D  
Figure 1: Current noise measurement system. feedback resistance, causing a large DC component. Higher values of resistance \(R_{R}\) will result in higher gain, but also in a reduced maximum bias current for the DUT. It will be assumed, however, that the gain of the first stage is sufficient to ensure a negligible influence of the noise contributions from the following stages to the overall noise of the system. The high-pass filter \(C_{1}R_{1}\) rejects the DC component at the output of \(OA_{1}\) so that only the noise signal can be further amplified by the voltage amplifier. The value of \(A_{v}\) ensures that the signal level at its output is compatible with the input range of the spectrum analyzer used for spectral estimation. The relevant noise sources that contribute to the background noise (BN) are also shown in Figure 1. For the amplifier to be effective, we need the noise contribution of these sources to be negligible with respect to the noise generated by the DUT. In the measurement bandwidth (determined by the corner frequency of the high-pass filter \(C_{1}R_{1}\) and the bandwidth limit of the transresistance and the voltage amplifier stages), the Power Spectral Density (PSD) \(S_{OUT}\) recorded by the spectrum analyzer is:  
\[S_{OUT}=(S_{ID}+S_{iBN})A_{R}^{2},\quad A_{R}=R_{R}A_{V} \tag{2}\]  
with  
\[S_{iBN}=S_{in}+\frac{S_{en}}{\left|Z_{D}\right\|R_{R}\right|^{2}}+\frac{S_{eB} }{\left|Z_{D}\right\|R_{R}\right|^{2}}+\frac{4 kT}{R_{R}} \tag{3}\]  
where \(S_{in}\), \(S_{en}\), and \(S_{eB}\) are the PSDs of the noise sources \(i_{n}\), \(e_{n}\) and \(e_{B}\), \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature. To obtain the expression of the background noise \(S_{iBN}\), as shown in Equation (3), we assumed all noise sources to be uncorrelated. When dealing with high DUT impedances, relatively high values of \(S_{en}\) and \(S_{eB}\) can be tolerated, and this means that Junction Field-Effect Transistor (JFET) or Metal-Oxide-Semiconductor Field Effect Transistor (MOSFET) input operational amplifiers can be used to minimize the contribution from the current noise \(S_{in}\). Indeed, in many cases of interest, the main contribution to the background noise comes from the feedback resistance \(R_{R}\), which must be chosen to be as large as possible, in a manner compatible with the limitations mentioned above as well as the desired bandwidth of the system [20,21]. However, for DUT impedances on the order of 1 k\(\Omega\) or less, the contribution from \(S_{en}\) and \(S_{eB}\) can become relevant and cannot be neglected any longer. To simplify this discussion, we limited our analyses to the low frequency range, in which the flicker noise component can be more easily detected. Moreover, at low frequencies, we assume the DUT impedance to behave as a resistance, that is \(Z_{D}\approx R_{D}\). For this reason, the DUT model does not contain any reactive component. Flicker noise can only be measured when it is greater than the thermal noise of the DUT. Therefore, the thermal noise of the DUT represents a reference level with respect to which the background noise of the amplifier must be minimized.