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A Comprehensive Overview of the Temperature-Dependent Modeling of the High-Power GaN HEMT Technology Using mm-Wave Scattering Parameter Measurements

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## Abstract

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## 1. Introduction

_{max}). Due to the lack of a native GaN substrate, GaN HEMT transistors are commonly realized using various foreign substrates, such as sapphire, silicon (Si), and silicon carbide (SiC). Compared to sapphire and Si, SiC is more expensive, but its excellent thermal conductivity makes it the material of choice for enhancing the power handling capability of GaN HEMTs. An efficient heat spreading is a mandatory requirement, especially when considering a large device operating at high power and high temperature, as in the present study. The S-parameters of the DUT were measured with a frequency range spanning from 200 MHz to 65 GHz and at four values of the ambient temperatures (T

_{a}), namely 35 °C, 90 °C, 145 °C, and 200 °C. The importance of investigating the impact of the temperature on the behavior of the GaN HEMTs lies in the fact that the device performance can considerably vary with the selected working thermal condition [56,57,58,59,60]. The bias point was fixed at V

_{DS}= 30 V and V

_{GS}= −3.1 V, corresponding to a dissipated power of 5.1 W when T

_{a}is set to 35 °C. To model the measured S-parameters, three different strategies are considered here: the equivalent-circuit approach, ANNs, and GRUs. The models are compared in terms of accuracy, generalizability, complexity, compactness, and usefulness. The equivalent circuit is a physically meaningful representation, enabling the achievement of deeper insights into the device physics, and is suitable to be used as a starting point for building both large-signal and noise models, which play a crucial role in a successful design of high power amplifiers (HPAs) and low noise amplifiers (LNAs). ANNs have good learning ability and generalization capabilities, allowing for achieving accurate predictions even for data different from those used in training. The approach based on GRUs allows for accurately fitting the small-signal behavior of the DUT, thus achieving a faithful reproduction of the experiments.

## 2. DUT and Modelling Methodologies

_{T}of 25 GHz, was optimized for X-band (i.e., 8–12 GHz) high-power applications. The S-parameters were measured with a frequency range spanning from 200 MHz to 65 GHz with a 200 MHz step at different T

_{a}, namely 35 °C, 90 °C, 145 °C, and 200 °C. In order to move the reference planes from the instrument ports to the probe tips, an off-wafer calibration was performed with line-reflect-match standards on a commercial impedance standard substrate (i.e., a GGB Industries CS-5 ISS). The temperature-dependent experiments were carried out without any heat sink, and a software-controlled thermal chuck was used to set the die temperature. The comparative analysis was performed by considering the bias point given by V

_{DS}= 30 V and V

_{GS}= −3.1 V. This operating bias condition implies a dissipated power of 5.1 W when considering the ambient temperature of 35 °C.

_{DS}= 0 V and V

_{GS}= −4 V), and eight intrinsic elements, which are bias dependent and are evaluated from the intrinsic admittance (Y-) parameters at the bias point of interest. At such a bias condition, the complexity of the equivalent circuit can be considerably reduced, thereby allowing extraction of the extrinsic equivalent-circuit elements. After applying the de-embedding of the eight extrinsic equivalent-circuit elements by using simple matrix manipulations, the intrinsic Y-parameters were calculated from the corresponding S-parameters by using the well-known conversion formulas, thereby allowing for the calculation of the intrinsic equivalent-circuit elements. MATLAB software environment is used for implementing the extraction of the equivalent-circuit elements from S-parameter measurements.

_{12}because of its more complex behavior that was not possible to reproduce accurately by using a single ANN for both real and imaginary parts. The training data are all the available measurements, with the exception of those at 145 °C, which are used for assessing the model generalization ability.

## 3. Experimental Results and Discussion

_{22}) is affected by the kink effect, which consists of an abrupt change in the behavior of this parameter at a certain frequency [63,64,65,66,67]. The GaN HEMT technology is prone to the kink effect in S

_{22}due to the relatively high transconductance (g

_{m}) [66]. All three investigated models allow for accurately reproducing the measurements over the full studied frequency range and under the four different ambient temperature conditions.

_{ij}) between measurements and simulations. The values of E

_{ij}are determined using the following formula with a total number of frequency points (N

_{f}) set to 325:

_{TOT}is evaluated by averaging the four achieved values of E

_{ij}between measurements and simulations of S

_{11}, S

_{21}, S

_{12}, and S

_{22}:

_{11}, E

_{12}, E

_{21}, E

_{22}, and E

_{TOT}for all three models at the four different temperature conditions. It was found that the two behavioral models based on using ANNs and GRUs allow reaching much lower percentage errors with respect to the equivalent-circuit model. In addition, it should be underlined that the ANNs enable achieving percentage errors similar to the ones of the GRUs even for data different from those used in training (i.e., 145 °C). This clearly proves the good generalization capability of the developed ANN-based modeling approach.

_{a}on the transconductance (g

_{m}), the total gate capacitance (C

_{gg}), and the intrinsic unity current-gain cut-off frequency (f

_{T}), which is determined as the ratio between g

_{m}and 2πC

_{gg}. As can be observed, when heating the DUT, the degradation of the electron transport properties causes a substantial reduction in g

_{m}and, in turn, of f

_{T}, which are crucial figures of merit for the assessment of the transistor performance. The obtained degradation of g

_{m}with increasing T

_{a}can be foreseen in the experiments by the observed reduction in the magnitude of the low-frequency forward transmission coefficient (S

_{21}) when heating the DUT (i.e., the magnitude of the measured S

_{21}at 200 MHz decreases from 22.57 to 13.84 by increasing T

_{a}from 35 °C to 200 °C).

## 4. Conclusions

^{2}GaN HEMT on a silicon carbide substrate. The tested device was heated up to 200 °C and biased at V

_{DS}= 30 V and V

_{GS}= −3.1 V. Three different modeling strategies were analyzed: an equivalent-circuit model, artificial neural networks, and gate recurrent units. It was found that each modeling strategy possesses its own inherent strengths and weaknesses, and the best choice depends on the actual use of the model. To be sure that the selected model is the best choice, it is necessary to carefully examine the model performance in terms of prediction accuracy, generalization ability, computational efficiency, representation compactness, and model usefulness. The developed analysis showed that the equivalent circuit is a compact representation based on using sixteen parameters, which allowed for achieving a good prediction accuracy and may be very useful as physically meaningful feedback for improving device fabrication and as a starting point for building both large-signal and noise models. On the contrary, behavioral-based techniques have the great advantage of allowing achieving straightforward better prediction accuracy without needing to determine an equivalent-circuit representation. In addition, ANNs allow for obtaining a very good prediction accuracy even for data different from those used in training, thereby demonstrating the model generalization ability of the behavioral-based approach.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Illustration of the model based on using artificial neural networks for the tested GaN HEMT: (

**a**) S

_{11}, S

_{21}, S

_{22}, and (

**b**) S

_{12}.

**Figure 5.**Illustration of the internal architecture of a GRU unit. The symbols “$\u2a01$” and “$\u2a02$” represent the operation of addition and element-wise multiplication, respectively.

**Figure 6.**Measured (blue symbols) and equivalent-circuit simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 35 °C.

**Figure 7.**Measured (blue symbols) and equivalent-circuit simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 90 °C.

**Figure 8.**Measured (blue symbols) and equivalent-circuit simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 145 °C.

**Figure 9.**Measured (blue symbols) and equivalent-circuit simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 200 °C.

**Figure 10.**Measured (blue symbols) and ANN simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 35 °C.

**Figure 11.**Measured (blue symbols) and ANN simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 90 °C.

**Figure 12.**Measured (blue symbols) and ANN simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 145 °C.

**Figure 13.**Measured (blue symbols) and ANN simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 200 °C.

**Figure 14.**Measured (blue symbols) and GRU simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 35 °C.

**Figure 15.**Measured (blue symbols) and GRU simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 90 °C.

**Figure 16.**Measured (blue symbols) and GRU simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 145 °C.

**Figure 17.**Measured (blue symbols) and GRU simulated (red lines) behavior of (

**a**) S

_{11}, (

**b**) S

_{12}, (

**c**) S

_{21}, and (

**d**) S

_{22}with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V, V

_{GS}= −3.1 V, and T

_{a}= 200 °C.

**Table 1.**Structure of the trained artificial neural networks: the number of neurons for the input, first hidden, second hidden, and output layers.

Parameter | ANN Structure |
---|---|

S_{11} | 2-4-4-2 |

S_{21} | 2-4-4-2 |

Re (S_{12}) | 2-5-5-1 |

Im (S_{12}) | 2-5-4-1 |

S_{22} | 2-5-5-2 |

Parameter | GRU Structure |
---|---|

Re (S_{ij}) | 5-1 |

Im (S_{ij}) | 5-1 |

**Table 3.**Percentage errors between measured and modeled scattering parameters with the frequency range spanning from 200 MHz to 65 GHz for the DUT at V

_{DS}= 30 V and V

_{GS}= −3.1 V, with four different values of T

_{a}. Three models are considered: Equivalent circuit, ANNs, and GRUs.

T_{a} (°C) | Parameter | Equivalent Circuit | ANNs | GRUs |
---|---|---|---|---|

35 | E_{11} | 45.96% | 1.04% | 1.32% |

E_{21} | 26.86% | 12.93% | 14.63% | |

E_{12} | 38.51% | 1.96% | 3.47% | |

E_{22} | 44.32% | 0.79% | 0.98% | |

E_{TOT} | 38.91% | 4.18% | 5.10% | |

90 | E_{11} | 43.31% | 1.14% | 1.44% |

E_{21} | 28.65% | 12.71% | 10.73% | |

E_{12} | 41.38% | 2.04% | 2.69% | |

E_{22} | 45.29% | 0.78% | 0.96% | |

E_{TOT} | 39.65% | 4.17% | 3.96% | |

145 | E_{11} | 38.62% | 2.12% | 2.10% |

E_{21} | 27.30% | 16.51% | 19.19% | |

E_{12} | 40.18% | 2.69% | 3.36% | |

E_{22} | 41.69% | 1.44% | 1.68% | |

E_{TOT} | 36.95% | 5.69% | 6.58% | |

200 | E_{11} | 35.96% | 1.64% | 1.69% |

E_{21} | 24.44% | 21.27% | 9.26% | |

E_{12} | 38.10% | 1.91% | 2.93% | |

E_{22} | 37.88% | 1.37% | 1.34% | |

E_{TOT} | 34.10% | 6.55% | 3.81% |

**Table 4.**Values of g

_{m}, C

_{gg}, and f

_{T}as a function of T

_{a}for a GaN HEMT at V

_{DS}= 30 V and V

_{GS}= −3.1 V.

T_{a} (°C) | g_{m} (mS) | C_{gg} (pF) | f_{T} (GHz) |
---|---|---|---|

35 | 347.9 | 2.560 | 21.63 |

90 | 293.6 | 2.372 | 19.70 |

145 | 249.7 | 2.222 | 17.89 |

200 | 215.4 | 2.130 | 16.09 |

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## Share and Cite

**MDPI and ACS Style**

Crupi, G.; Latino, M.; Gugliandolo, G.; Marinković, Z.; Cai, J.; Bosi, G.; Raffo, A.; Fazio, E.; Donato, N.
A Comprehensive Overview of the Temperature-Dependent Modeling of the High-Power GaN HEMT Technology Using mm-Wave Scattering Parameter Measurements

. *Electronics* **2023**, *12*, 1771.
https://doi.org/10.3390/electronics12081771

**AMA Style**

Crupi G, Latino M, Gugliandolo G, Marinković Z, Cai J, Bosi G, Raffo A, Fazio E, Donato N.
A Comprehensive Overview of the Temperature-Dependent Modeling of the High-Power GaN HEMT Technology Using mm-Wave Scattering Parameter Measurements

. *Electronics*. 2023; 12(8):1771.
https://doi.org/10.3390/electronics12081771

**Chicago/Turabian Style**

Crupi, Giovanni, Mariangela Latino, Giovanni Gugliandolo, Zlatica Marinković, Jialin Cai, Gianni Bosi, Antonio Raffo, Enza Fazio, and Nicola Donato.
2023. "A Comprehensive Overview of the Temperature-Dependent Modeling of the High-Power GaN HEMT Technology Using mm-Wave Scattering Parameter Measurements

" *Electronics* 12, no. 8: 1771.
https://doi.org/10.3390/electronics12081771