## 2 Materials and Methods  
The reliability of GaN HEMT is superior than Si MOSFET in a neutron radiation environment as proven experimentally in our previous work in [36]. Thus, one can assure the reliability of the GaN Operational Amplifier (OPAMP) designed in this work. In this work, we chose GaN HEMT transistors from efficient power conversion (EPC) in order to design the operational amplifier and readout circuit because they show better performance in a radiation environment, as reported in the literature, where negligible variations in the GaN HEMT parameters were reported in a fast neutron ambient (up to a fluence of \(1\times 10^{15}\)) [37; 38; 39].  
OPAs are the most used devices for designing the various modules of the readout system as shown in Figure 1 which is modified from reference [9], including transimpedance amplifiers for current to voltage conversion, adders for signal regrouping and integrators for charge to time conversion. The operational amplifier required to design these circuits should have high gain, higher bandwidth, and high speed so that it can process the short-rise time and weak signals obtained from the photomultiplier tube (PMT). The first block of the system is a scintillator crystal along with PSPMT for detecting the prompt signal and generating the corresponding current signal. The PSPMT has 64 readout channels which is reduced to 4 channels (I1, I2, I3, and I4) by the discretized position circuit (DPC) proposed in [40]. The four current signals from the PSPMT are then converted into four voltages (V1, V2, V3, and V4) by four transimpedance amplifiers (TIAs). The output voltage of the TIA is proportional to the input current i.e.,  
\[V=I\cdot R \tag{1}\]  
where \(I\) is the input current to the TIA and \(R\) is the feedback resistance in the TIA circuit. The signals obtained from the TIAs are then fed to the signal regrouping block where they are combined to provide the following:  
\[V_{S1}=V1+V2+V3+V4 \tag{2}\]  
\[V_{S2}=V1+V2-V3-V4 \tag{3}\]  
\[V_{S3}=V1+V4-V2-V3 \tag{4}\]  
The gain of the adder circuit in the signal regrouping block is set to unity so that Equations (2)-(4) can be obtained in terms of the current from the PMT as follows:  
\[V_{S1}=(I1+I2+I3+I4)\cdot R \tag{5}\]  
\[V_{S2}=(I1+I2-I3-I4)\cdot R \tag{6}\]  
\[V_{S3}=(I1+I4-I2-I3)\cdot R \tag{7}\]  
The output signal \(V_{S1}\) is fed to the integrator and upon integration, the total charge accumulated at the detector can be obtained which provides the energy information of the radiation event. The output signals \(V_{S1}\), \(V_{S2}\), \(V_{S3}\), and VIE (output of the integrator) are then sent for data processing for computing the energy and position information of the radiation event. To prevent the pulse pile-up, the output of the integrator is reset to the initial state after each detection by the reset signal provided by the data processing unit. The details of the energy and position calculation are discussed in the next section.  
Figure 2 shows the schematic diagram of the proposed operational amplifier. The first stage of the circuit is a dual input balanced differential amplifier formed by M1 and M2. Since a complimentary GaN transistor is not available, passive load is used throughout the design. Transistors M3, M4, M7, M9 and resistor R3 are in current mirror configuration to provide the tail current of the first, second and third stage of the circuit. The second stage is the dual input unbalanced output differential amplifier formed by M5 and M6. M10 and M11 are configured to a common source with degeneration mode and cascaded together to achieve high open-loop gain. M8 is used in common drain configuration between the second stage and the common source stage to prevent loading. Capacitor C1 is added to improve the stability of the circuit. LTSpice is employed for the design and simulation of the OPA by importing the Simulation Program with Integrated Circuit Emphasis (SPICE) model of GaN HEMT provided by EPC.