## 3 Results and Discussion
### 1/f (Low-Frequency) Noise  
To gain insights into the primary locations of the trapping phenomena resulting from the defect site in response to the fluctuation of the channel carriers, \(1/f\) noise measurements were conducted [32]. This is an effective measurement method for analyzing interfaces that can cause device performance degradation. This measurement was performed using a low-noise current amplifier in conjunction with a signal analyzer. Frequencies up to \(10^{4}\) Hz were covered at \(V_{DS}\) = 0.05 V. Figure 4 shows the normalized drain current noise (\(S_{ld}/I_{D}{}^{2}\)) for \(I_{D}\) = 1 A/\(\upmu\)m. A lower \(1/f\) noise characteristic indicates fewer traps or defects. LM-HEMTs had a lower drain current noise density than MHEMT. The frequency dependence of \(S_{ld}/I_{D}{}^{2}\), also known as flicker noise or \(1/f\) noise, can be described using a power-law equation of \(1/f\)\({}^{\gamma}\). In the given frequency range, the noise data were fitted using the \(1/f\)\({}^{\gamma}\) function, and the frequency exponent (\(\gamma\)) was determined. The value of \(\gamma\) provides crucial information about the nature of the trapping mechanisms and their influence on the noise characteristics in the devices [33; 34]. The frequency exponent being close to 1 in \(1/f\) noise measurements typically suggests that the dominant traps responsible for the noise are nearly constant in terms of energy and depth [28]. This behavior indicates that these traps have a relatively uniform impact on the noise characteristics across different frequencies, leading to the observed power-law behavior in the \(1/f\) noise spectrum. [33; 34]. A value of \(\gamma\) less than 1 for the MHEMT suggests that the primary trapping site is situated near the bulk traps, which are positioned away from the channel interface. On the other hand, the LM-HEMTs with a \(\gamma\) greater than 1 indicate that the location of the dominant trapping phenomena is situated at the interface of the InGaAs channel. The difference in \(\gamma\) values provides valuable insights into the nature and locations of the dominant traps contributing to the \(1/f\) noise in each type of device. There are several physical theories that provide explanations of \(1/f\) noise [35; 36]. According to the McWhorter model, the initial source of \(1/f\) is the channel's fluctuating carrier number as a result of carrier trapping and de-trapping. On the other hand, the Hooge model suggests that noise results from carriers' fluctuating mobility caused by phonon-scattered carriers. In the case of the device used in this study, it was determined that the device's noise characteristics followed the McWhorter model. So, to achieve a quantitative comparison, we derived \(N_{t}\) utilizing the carrier-number-fluctuation (CNF) model [37; 38; 39; 31]:  
\[\frac{S_{ld}}{I_{D}^{2}}=\left(\frac{g_{m}}{I_{D}}\right)^{2}S_{Vfb} \tag{2}\]  
Figure 4: Comparison of the normalized drain current noise spectral density (\(S_{ld}/I_{D}{}^{2}\)) at \(V_{DS}\) = 0.05 V and \(I_{D}\) = 1 A/\(\upmu\)m.  
\[\text{with }S_{Vfb}=\frac{q^{2}N_{k}kT\lambda}{WLC_{ins}^{2}f} \tag{3}\]  
\(S_{Vfb}\) represents the flat band power spectral density, \(kT\) stands for the thermal energy, \(WL\) corresponds to the channel area, \(C_{ins}\) signifies the insulator capacitance,\(f\) represents the frequency, and \(N_{t}\) denotes the bulk trap density. In addition, \(\lambda\) is the tunneling attenuation distance and can be calculated using the formula \(\lambda\) = [4\(\pi\)(2\(m\)\(\ast\)\(\phi_{B}\))\({}^{1/2}\)/\(h\)]\({}^{-1}\), where \(\Phi_{B}\) stands for the barrier height and \(m\)\({}^{\ast}\) denotes the effective mass. In the CNF model, both the terms \(S_{ld}\)/\(I_{D}\)\({}^{2}\) and (\(g_{m}\)/\(I_{D}\))\({}^{2}\) exhibit similar variations across a range of drain currents or gate voltages. As depicted in Figure 5a,b, both \(S_{ld}\)/\(I_{D}\)\({}^{2}\) and (\(g_{m}\)/\(I_{D}\))\({}^{2}\) display comparable variations over multiple decades as the drain current changes.