# TCAD Modeling of GaN HEMT Output Admittance Dispersion through Trap Rate Equation Green’s Functions

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## Abstract

**:**

## 1. Introduction

## 2. Trap Dynamic Model Implementation

- Analysis of nonlinear dynamic operation, and in particular large-signal periodic (in our implementation through the Harmonic Balance method);
- Straightforward extension to non-local trap mechanisms;
- Availability of the Green’s Functions of the trap rate equations, that are useful for numerically efficient sensitivity analysis.

- The system (11) is converted into the frequency domain, yielding the HB system [22]$${\mathbf{D}}^{\left(\alpha \right)}\mathsf{\Omega}\mathbf{X}=\mathsf{\Gamma}{\mathbf{F}}^{\left(\alpha \right)}({\mathsf{\Gamma}}^{-1}\mathbf{X},\mathbf{E};\mathbf{\sigma})\phantom{\rule{2.em}{0ex}}\alpha =\phi ,n,p,{n}_{T}$$$${Y}_{q,r}=\frac{{\displaystyle {I}_{1}^{\left(q\right)}}}{{\displaystyle {V}_{1}^{\left(r\right)}}}$$
- System (12) is linearized and the CGFs are computed with negligible numerical effort using the algorithms detailed in [22,28]. Furthermore, assuming a static parameter variation $\Delta \mathbf{\sigma}$, the microscopic local sources ${\mathbf{S}}_{\alpha ,l}\left(\mathbf{r}\right)$ are computed as the residual of (12) evaluated with the nominal solution and $\mathbf{\sigma}={\mathbf{\sigma}}_{0}+\Delta \mathbf{\sigma}$ [20]. Since the parameter variation is static, the varied residual is characterized by the same spectrum of the nominal solution, as shown in Figure 1 (middle).
- The variation of the AC terminal currents due to $\Delta \mathbf{\sigma}$ is recovered by the convolution integral$$\Delta {I}_{1}^{\left(q\right)}=\sum _{\alpha}{\int}_{\mathsf{\Omega}}\phantom{\rule{0.277778em}{0ex}}\sum _{l=-1,0,+1}{\left({\mathbf{G}}_{\alpha}^{\left(q\right)}\left(\mathbf{r}\right)\right)}_{(1,l)}{\mathbf{S}}_{\alpha ,l}\left(\mathbf{r}\right)\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}\mathbf{r}$$A graphic interpretation of the convolution is sketched in Figure 1 (right), where it is evident that (14) can be split into three individual contributions accounting for the frequency conversion effects towards the fundamental frequency. In particular, the $(1,0)$ element of the Green’s Function accounts for the conversion from DC.

## 3. HEMT Device Structure

## 4. Sensitivity of Real $\mathbf{\left(}{\mathit{Y}}_{\mathbf{DD}}\mathbf{\right)}$

## 5. Sensitivity of $\mathbf{Imag}\mathbf{\left(}{\mathit{Y}}_{\mathbf{DD}}\mathbf{\right)}$

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Mishra, U.; Likun, S.; Kazior, T.; Wu, Y.F. GaN-Based RF Power Devices and Amplifiers. Proc. IEEE
**2008**, 96, 287–305. [Google Scholar] [CrossRef] - Meneghesso, G.; Verzellesi, G.; Danesin, F.; Rampazzo, F.; Zanon, F.; Tazzoli, A.; Meneghini, M.; Zanoni, E. Reliability of GaN High-Electron-Mobility Transistors: State of the Art and Perspectives. IEEE Trans. Device Mater. Reliab.
**2008**, 8, 332–343. [Google Scholar] [CrossRef] - UMS GaN GH15-10. Available online: https://www.ums-rf.com/ums-gan-gh15-10-technology-is-space-evaluated/ (accessed on 28 May 2023).
- The Qorvo Gan Advantage. Available online: https://www.qorvo.com/innovation/technology/gan (accessed on 28 May 2023).
- MMIC Advanced Technology. Available online: https://www.winfoundry.com/en-US/Tech/tech_advanced (accessed on 28 May 2023).
- III—V Processes. Available online: https://www.ommic.com/iii-v-processes/ (accessed on 28 May 2023).
- Wang, K.; Jiang, H.; Liao, Y.; Xu, Y.; Yan, F.; Ji, X. Degradation Prediction of GaN HEMTs under Hot-Electron Stress Based on ML-TCAD Approach. Electronics
**2022**, 11, 3582. [Google Scholar] [CrossRef] - Modolo, N.; Santi, C.D.; Minetto, A.; Sayadi, L.; Prechtl, G.; Meneghesso, G.; Zanoni, E.; Meneghini, M. Trap-state mapping to model GaN transistors dynamic performance. Sci. Rep.
**2022**, 12, 1755. [Google Scholar] [CrossRef] [PubMed] - Joshi, V.; Soni, A.; Tiwari, S.P.; Shrivastava, M. A Comprehensive Computational Modeling Approach for AlGaN/GaN HEMTs. IEEE Trans. Nanotechnol.
**2016**, 15, 947–955. [Google Scholar] [CrossRef] - Zagni, N.; Verzellesi, G.; Chini, A. Temperature-Independent Current Dispersion in 0.15 μm AlGaN/GaN HEMTs for 5G Applications. Micromachines
**2022**, 13, 2244. [Google Scholar] [CrossRef] [PubMed] - Angelotti, A.M.; Gibiino, G.P.; Santarelli, A.; Florian, C. Experimental Characterization of Charge Trapping Dynamics in 100-nm AlN/GaN/AlGaN-on-Si HEMTs by Wideband Transient Measurements. IEEE Trans. Electron Devices
**2020**, 67, 3069–3074. [Google Scholar] [CrossRef] - Oishi, T.; Otsuka, T.; Tabuchi, M.; Yamaguchi, Y.; Shinjo, S.; Yamanaka, K. Bias Dependence Model of Peak Frequency of GaN Trap in GaN HEMTs Using Low-Frequency Y22 Parameters. IEEE Trans. Electron Devices
**2021**, 68, 5565–5571. [Google Scholar] [CrossRef] - Vertiatchikh, A.; Eastman, L. Effect of the surface and barrier defects on the AlGaN/GaN HEMT low-frequency noise performance. IEEE Electron Device Lett.
**2003**, 24, 535–537. [Google Scholar] [CrossRef] - Beleniotis, P.; Schnieder, F.; Chevtchenko, S.; Rudolph, M. Localization of Trapping Effects in GaN HEMTs with Pulsed S-parameters and Compact Models. In Proceedings of the 2022 17th European Microwave Integrated Circuits Conference (EuMIC), Milan, Italy, 26–27 September 2022; IEEE: Piscataway, NJ, USA, 2022. [Google Scholar] [CrossRef]
- Gomes, J.L.; Nunes, L.C.; Pedro, J.C. Transient Pulsed S-Parameters for Trapping Characterization. In Proceedings of the 2020 International Workshop on Integrated Nonlinear Microwave and Millimetre-Wave Circuits (INMMiC), Cardiff, UK, 16–17 July 2020; IEEE: Piscataway, NJ, USA, 2020. [Google Scholar] [CrossRef]
- Nicollian, E.H.; Goetzberger, A. MOS Conductance Technique For Measuring Surface State Parameters. Appl. Phys. Lett.
**1965**, 7, 216–219. [Google Scholar] [CrossRef] - Nicollian, E.; Goetzberger, A.; Lopez, A. Expedient method of obtaining interface state properties from MIS conductance measurements. Solid-State Electron.
**1969**, 12, 937–944. [Google Scholar] [CrossRef] - Subramani, N.K. Physics-Based TCAD Device Simulations and Measurements of GaN HEMT Technology for RF Power Amplifier Applications. Ph.D. Dissertation, University of Limoges, Limoges, France, 2017. Available online: https://tel.archives-ouvertes.fr/tel-01702325 (accessed on 28 May 2023).
- Shanbhag, A.; Sruthi, M.P.; Chakravorty, A.; DasGupta, N.; DasGupta, A. Compact Modeling of Static and Transient Effects of Buffer Traps in GaN HEMTs. IEEE Trans. Electron Devices
**2022**, 69, 999–1005. [Google Scholar] [CrossRef] - Donati Guerrieri, S.; Bonani, F.; Bertazzi, F.; Ghione, G. A Unified Approach to the Sensitivity and Variability Physics-Based Modeling of Semiconductor Devices Operated in Dynamic Conditions—Part I: Large-Signal Sensitivity. IEEE Trans. Electron Devices
**2016**, 63, 1195–1201. [Google Scholar] [CrossRef] - Donati Guerrieri, S.; Bonani, F.; Bertazzi, F.; Ghione, G. A Unified Approach to the Sensitivity and Variability Physics-Based Modeling of Semiconductor Devices Operated in Dynamic Conditions.—Part II: Small-signal and Conversion Matrix Sensitivity. IEEE Trans. Electron Devices
**2016**, 63, 1202–1208. [Google Scholar] [CrossRef] - Bonani, F.; Donati Guerrieri, S.; Ghione, G.; Pirola, M. A TCAD approach to the physics-based modeling of frequency conversion and noise in semiconductor devices under large-signal forced operation. IEEE Trans. Electron Devices
**2001**, 48, 966–977. [Google Scholar] [CrossRef] - Raja, P.V.; Subramani, N.K.; Gaillard, F.; Bouslama, M.; Sommet, R.; Nallatamby, J.C. Identification of Buffer and Surface Traps in Fe-Doped AlGaN/GaN HEMTs Using Y21 Frequency Dispersion Properties. Electronics
**2021**, 10, 3096. [Google Scholar] [CrossRef] - Potier, C.; Martin, A.; Campovecchio, M.; Laurent, S.; Quere, R.; Jacquet, J.C.; Jardel, O.; Piotrowicz, S.; Delage, S. Trap characterization of microwave GaN HEMTs based on frequency dispersion of the output-admittance. In Proceedings of the 2014 9th European Microwave Integrated Circuit Conference, Rome, Italy, 6–7 October 2014; IEEE: Piscataway, NJ, USA, 2014. [Google Scholar] [CrossRef]
- Synopsys Sentaurus. Available online: https://www.synopsys.com/silicon/tcad/device-simulation/sentaurus-device.html (accessed on 28 May 2023).
- Bonani, F.; Ghione, G. Noise in Semiconductor Devices; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Troyanovsky, B.; Yu, Z.; Dutton, R.W. Large Signal Frequency Domain Device Analysis Via the Harmonic Balance Technique. In Simulation of Semiconductor Devices and Processes; Springer: Vienna, Austria, 1995; pp. 114–117. [Google Scholar] [CrossRef]
- Donati Guerrieri, S.; Pirola, M.; Bonani, F. Concurrent Efficient Evaluation of Small-Change Parameters and Green’s Functions for TCAD Device Noise and Variability Analysis. IEEE Trans. Electron Devices
**2017**, 64, 1269–1275. [Google Scholar] [CrossRef] - Golio, J.; Miller, M.; Maracas, G.; Johnson, D. Frequency-dependent electrical characteristics of GaAs MESFETs. IEEE Trans. Electron Devices
**1990**, 37, 1217–1227. [Google Scholar] [CrossRef] - Donati Guerrieri, S.; Ramella, C.; Catoggio, E.; Bonani, F. Bridging the Gap between Physical and Circuit Analysis for Variability-Aware Microwave Design: Modeling Approaches. Electronics
**2022**, 11, 860. [Google Scholar] [CrossRef]

**Figure 1.**Graphic representation of the sensitivity analysis through Conversion Green’s Functions. Left: step 1. Middle: step 2. Right: step 3.

**Figure 3.**(

**Left**) static (solid) and pulsed (dotted) output characteristics. Black lines correspond to ${V}_{\mathrm{G}}$ from varying from −2.5 V to 0 V with 0.5 V step. The red line corresponds to ${V}_{\mathrm{G}}=-2.2$ V. The quiescent bias point used for pulsed simulations is shown by the red dot. (

**Right**) occupied trap density in the bias point.

**Figure 4.**Real (

**left**) and Imaginary (

**right**) part of ${Y}_{\mathrm{DD}}$ with nominal energy ${E}_{T}={E}_{C}-0.45$ eV and bias point shown in Figure 3.

**Figure 5.**Real part of ${Y}_{\mathrm{DD}}$ with varying trap energy levels. Lines: AC analysis from (1)–(4). Symbols: GF approach.

**Figure 6.**(

**Left**) 3D plot of the real part of the distributed variation source ${K}_{{n}_{T}}^{(D,D)}\left(\mathbf{r}\right)$ at ${f}_{\mathrm{peak}}=2.15$ kHz. (

**Right**) Zoom on the dotted region under gate.

**Figure 7.**(

**Left**) 3D plot of the real part of the DC microscopic local source ${\mathbf{S}}_{{n}_{T},0}\left(\mathbf{r}\right)$ at ${f}_{\mathrm{peak}}=2.15$ kHz. (

**Right**) Zoom on the dotted region under gate.

**Figure 8.**(

**Left**) 3D plot of the (1,0) component of the trap rate equation Green’s Function ${\left({\mathbf{G}}_{{n}_{T}}^{\left(D\right)}\left(\mathbf{r}\right)\right)}_{(1,0)}$ (real part) at ${f}_{\mathrm{peak}}=2.15$ kHz. (

**Right**) Zoom on the dotted region under gate.

**Figure 9.**Imaginary part of ${Y}_{\mathrm{DD}}$ at different trap levels. Lines: INC approach. Symbols: GF approach.

**Figure 10.**(

**Left**) 3D plot of the imaginary part of distributed variation source ${K}_{{n}_{T}}^{(D,D)}\left(\mathbf{r}\right)$ at ${f}_{1}=464$ Hz. (

**Right**) Zoom on the dotted region under gate.

**Figure 11.**(

**Left**) 3D plot of the imaginary part of the distributed variation source ${K}_{{n}_{T}}^{(D,D)}\left(\mathbf{r}\right)$ at ${f}_{2}=10$ kHz. (

**Right**) Zoom on the dotted region under gate.

**Figure 12.**2D plot of the (1,0) component of the trap rate equation Green’s Function ${\left({\mathbf{G}}_{{n}_{T}}^{\left(D\right)}\left(\mathbf{r}\right)\right)}_{(1,0)}$ (imaginary part). (

**Left**) ${f}_{1}=464$ Hz. (

**Right**) ${f}_{2}=10$ kHz.

Real (${\mathit{Y}}_{\mathit{DD}}$) [S/cm] | ${\mathit{R}}_{\mathit{out}}$ [$\mathsf{\Omega}$ cm] | |
---|---|---|

Static DC | $8.2\times {10}^{-3}$ | 122 |

AC $\mathrm{Real}\phantom{\rule{3.33333pt}{0ex}}\left({Y}_{\mathrm{DD}}\right)$ (low frequency) | $8.1\times {10}^{-3}$ | 124 |

Pulsed DC | 0.17 | 6 |

AC $\mathrm{Real}\phantom{\rule{3.33333pt}{0ex}}\left({Y}_{\mathrm{DD}}\right)$ (high frequency) | 0.17 | 6 |

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**MDPI and ACS Style**

Catoggio, E.; Donati Guerrieri, S.; Bonani, F.
TCAD Modeling of GaN HEMT Output Admittance Dispersion through Trap Rate Equation Green’s Functions. *Electronics* **2023**, *12*, 2457.
https://doi.org/10.3390/electronics12112457

**AMA Style**

Catoggio E, Donati Guerrieri S, Bonani F.
TCAD Modeling of GaN HEMT Output Admittance Dispersion through Trap Rate Equation Green’s Functions. *Electronics*. 2023; 12(11):2457.
https://doi.org/10.3390/electronics12112457

**Chicago/Turabian Style**

Catoggio, Eva, Simona Donati Guerrieri, and Fabrizio Bonani.
2023. "TCAD Modeling of GaN HEMT Output Admittance Dispersion through Trap Rate Equation Green’s Functions" *Electronics* 12, no. 11: 2457.
https://doi.org/10.3390/electronics12112457