A Systematic and Numerical Methodology for GaN HEMT Current-Gain Peak Analysis Using the Complex Lorentzian Function

The purpose of this letter is to present a measurement-based analysis of the transistor current-gain peak (CGP), which consists of a sudden peak in the magnitude of the short-circuit current-gain ( $h_{21}$ ) at a certain frequency. A systematic and numerical approach is proposed to analyze CGP. This powerful and technology-independent methodology is based on developing an accurate fitting of the experiments using the complex Lorentzian function, thus allowing an accurate and straightforward extraction of the parameters describing CGP. The validity of the developed technique is fully demonstrated by its application to the analysis of CGP for a gallium nitride (GaN) high-electron-mobility transistor (HEMT) at different ambient temperatures and bias conditions.


I. INTRODUCTION
T O MEET the increasing demand requirements of highfrequency and ultra-wideband applications, transistor technologies are aggressively pushed to perform faster. A wellknown figure-of-merit to quickly assess the high-frequency performance of a transistor is the unity current-gain cut-off frequency ( f T ), which is given by the frequency where the short-circuit current-gain (h 21 ) becomes unity. Though much of the interest in achieving an active transistor operation even at frequencies beyond f T has been directed toward bipolar transistors [1], [2], [3], [4], [5], [6], [7], [8], such as the so-called resonance phase transistor (RPT) [1], [2], [3], [4], [5], [6], recent efforts have been made to study the current-gain peak (CGP) for field-effect transistors (FETs) [9], [10], [11], [12], [13], [14], [15], [16], [17]. CGP consists of a peak easily detectable by plotting the magnitude of h 21 in decibels versus frequency on a log scale. CGP has been studied in terms of equivalent-circuit elements, and its appearance has been attributed to the resonance between the extrinsic inductances (i.e., the sum of L d and L s ) and the intrinsic capacitances (i.e., the parallel connection of C ds and the series connection of C gs and C gd ) [9]. CGP has been studied also in terms of poles and zeros, and a simplified mathematical expression has been proposed to calculate the location of the peak [14]. As CGP consists of an abrupt change in the magnitude of h 21 at a certain frequency, this anomalous peak can be seen as a kink effect affecting h 21 . Hence, analogously to what has been done for the kink effect in the output reflection coefficient (S 22 ) [18], CGP can be investigated in terms of the second derivative (D2) of the magnitude of h 21 in decibels versus frequency, enabling the determination of a set of parameters to fully and systematically characterize this kink effect. Due to the high sensitivity to noise of the second derivative, trustworthy values of the kink parameters cannot be calculated directly from measurements. Therefore, model simulations [e.g., equivalent circuits and artificial neural networks (ANNs)] or fitting functions are required to obtain a smooth behavior of D2. To reduce the modeling effort and to enable a more accurate reproduction of the experiments, scattering (S-) parameters have been straightforwardly modeled using ANNs [19] and then, CGP has been described with a set of kink parameters [11]. Recently, an alternative approach has been proposed to quantify CGP by measuring the area of CGP (ACGP) as the zone between the two curves corresponding to h 21 with and without the peak [15]. This task has been done by using a commercial plot digitizer without the need of determining a model able to approximate the measurements.
Here, we propose a novel methodology based on developing a fitting procedure using a complex Lorentzian function [20] to reproduce the frequency-dependent behavior of h 21 , thereby allowing an accurate determination of a set of kink parameters describing CGP systematically and numerically. The developed methodology is applied to the gallium nitride (GaN) technology, as its operating frequency keeps extending further in the mm-wave range [21], [22], [23], [24], [25]. The described procedure is technology independent, so, as a criterion of choice, we selected a device for which a complete measurement set was available. In particular, the studied device is a aluminium gallium nitride (AlGaN)/GaN highelectron-mobility transistor (HEMT) on SiC substrate with a gate length of 0.7 µm and a gate width of 2 × 400 µm. Experiments consist of multibias S-parameters measured from 300 MHz to 40 GHz with a frequency step of 198.5 MHz at five ambient temperatures (T a ): 20 • C, 35 • C, 50 • C, 65 • C, and 80 • C. The proposed methodology is used to fit h 21 versus frequency without the need of modeling all the four S-parameters and, subsequently, to determine the parameters describing CGP, thereby enabling the analysis of the effects of varying bias and temperature conditions.

II. FITTING PROCEDURE
The behavior of h 21 as a function of the frequency ( f ) has been modeled by using the following complex function: The first term is a complex Lorentzian function used to describe the resonant peak occurring in the high-frequency range [20], [26], [27]. f r is its resonant frequency, Q is the quality factor of the peak, S is a complex coefficient, and i is the imaginary unit. The second term (i.e., a/f ), where a is a complex coefficient, is used to model the response of h 21 at lower frequencies. Moreover, a complex linear function (i.e., bf + c) has been included to reduce the fitting residuals by modeling the background effects over the considered frequency range [28], [29], [30]. The fitting procedure has been implemented in Python using the lmfit library, and the optimized coefficients have been calculated by means of the Levenberg-Marquardt algorithm. A comparison between the measured and fitted h 21 for the whole studied frequency range is reported in Fig. 1(a). The coefficient of determination (i.e., R 2 ) has been estimated for each fitting. It is a goodness-of-fit statistic that describes how well the model explains the measurements carried out [31]. In this study, the R 2 is better than 0.99, indicating a strong correlation between the measured and fitted data. To better highlight the fitting performance in the region close to CGP, Fig. 1(b) shows the comparison for the limited frequency range from 15 to 40 GHz. Finally, to give an intuitive idea on how the three complex terms in (1) contribute to reproducing the behavior of h 21 , Fig. 1(c) shows the magnitude in decibels of h 21 together with the magnitude in decibels of the three terms.
In addition to Q-factor and f r , which are straightforwardly estimated from the complex Lorentzian function, and to the kink amplitude (KA), which is the value of the fitted h 21 at the resonance frequency, the kink parameters include the following ones evaluated from D2 (see Fig. 2).
1) The kink frequency band (KFB) is the frequency range going from the kink frequency start (KFS) to the kink frequency end (KFE), i.e., the frequencies of CGP onset and disappearance, which are defined as the two frequencies where D2 becomes 0.
2) The kink size (KS) represents the CGP size and is given by the negative peak value of D2 occurring at the kink frequency (KF).
3) The kink shape factor (KSF) is calculated as the ratio between KF and KFB.  occurrence of CGP and its bandwidth (i.e., Q-factor is defined as the ratio of f r to the 3-dB bandwidth and is estimated from the complex Lorentzian function, whereas the KSF is given by the ratio between KF and KFB that is calculated from D2 of the fitted h 21 ), Figs. 3(d) and 4(d) show a good agreement between these two parameters, confirming their validity.

III. EXPERIMENTAL RESULTS
As can be seen in Figs. 3(e) and 4(e), the parameters KA and the magnitude of KS, which are defined in different ways to quantify the size of CGP (i.e., KA and KS are estimated at the frequency occurrence of CGP from the fit h 21 and from its D2, respectively), show similar trends but with some discrepancies (e.g., Fig. 3(e) shows that, by heating the device at 80 • C, KA is reduced while |KS| is increased), demonstrating the need of defining KS for evaluating the level of enhancement of CGP as higher values of h 21 do not necessarily imply an enhanced CGP.
By heating the device, the most evident effect on h 21 is the reduction of its low-frequency magnitude [see Fig. 3(a)], due to the degradation of the carrier transport properties and, then, of the intrinsic transconductance (g m ) [32]. KF shows only a weak dependence on T a , consistent with the fact that the extrinsic inductances and the intrinsic capacitances depend only slightly on the temperature. Fig. 4(b) shows that variations in V DS have a very strong impact on CGP, which disappears at low V DS values because of the increase in the intrinsic output conductance (g ds ) which tends to short circuit the contributions of the intrinsic capacitances [9]. By lowering V DS , CGP becomes less pronounced (as can be quantified by the reduction in the magnitude of KS and in KA) and appears at low frequencies (as can be quantified by the reduction in KF, KFS, and KFE) and over a wider frequency range (as can be quantified by the broader KFB). In addition, the shift of CGP at low frequencies and over a broader frequency range leads to a reduction in both the Q-factor and KSF.

IV. CONCLUSION
A fitting procedure based on a complex Lorentzian function was developed and, then, successfully applied to mimic the frequency-dependent behavior of the transistor short-circuit current gain. The optimized coefficients were calculated using the Levenberg-Marquardt algorithm. The fit function was used for determining the parameters that allow one to fully characterize CGP systematically and numerically. Although a GaN HEMT at different T a and bias conditions was considered as a case study, the proposed methodology is general and technology independent and, then, applicable for investigating CGP for any type of transistor with respect to different factors.