Understanding the Thermal Time Constants of GaN HEMTs through Model Order Reduction Technique
Round 1
Reviewer 1 Report
Review Report on electronics-1477894
The MS entitled ” Understanding the Thermal Time Constants of GaN HEMTs through Model Order Reduction Technique” by A. Jakani and coworkers present an study on the thermal time constants of GaN using numerical methods.
The paper is well organized and its English is satisfactory enough.
The paper compares the result a numerical Finite Element Analysis (FEA) and an analytical approach to extract the thermal time constants and the thermal resistances of GaN structures. In the MS authors have extracted the time constants using a Model Order Reduction (MOR) technique based on the Ritz vector approach. They found same values for the time constants using the analytical model and the ones obtained through extraction the FEA method. The results presented in the paper certainly are of interest for researchers working in the field.
Nevertheless, while results are clearly enough to validate the MOR technique (that seems to be the main aim of the paper) I am not fully convinced about the claim of the authors about the use of the method in advanced (deep-submicrometer/nanometer scaled channels) of HEMTs.
It is well known in short channels momentum and energy exchange between carriers (charge transport) is not in equilibrium with lattice (hot carries, ballistic transport, …) and advanced models using the Monte Carlo technique were proposed years ago (see for instance: D. Lacroix, et al. Physical Review B, 72, 064305, 2005 for silicon and germanium) to replace the Fourier law that becomes invalid in short channel FETs.
1 – According to the above I request the introduction of a paragraph with a critical discussion about the validity of their model in terms of channel’s length and thickness and how – if possible- the model could be modified to account for the phenomena in the previous paragraph.
2 – Authors must also give a more detailed description of the model (equations and boundary conditions) used to simulate the GaN HEMT.
Author Response
Response to Reviewer 1 Comments
Point 1 – According to the above I request the introduction of a paragraph with a critical discussion about the validity of their model in terms of channel’s length and thickness and how – if possible- the model could be modified to account for the phenomena in the previous paragraph.
Response 1 : In all these approaches, only Fourier’s Law is considered. We are aware that Fourier approach may be limited for nanoscale device [9][10] but we are more focused on an approach that will address and improve thermal models for electronic industry beyond a simple RC cell for real devices. Nevertheless, 2 references have been added: one of Dr Cahill et al., one of D Lacroix et al.
These lines have been added.
Point 2 – Authors must also give a more detailed description of the model (equations and boundary conditions) used to simulate the GaN HEMT.
Response 2 : The thermal model is obtained from the Finite Element formulation of the classical heat equation based on the Fourier phenomenological approach. Assumptions are adiabatic conditions on lateral directions, and a uniform heat flux the surface representing the gate of the device, a fixed baseplate temperature.
These lines have been added.
Reviewer 2 Report
The paper describes the thermal time constants of GaN HEMTs transistor. It also shows the model order reduction. Thermal issue are shown. It compares analythical model with numerical simulation. It would be better if the authors shown measurement results of a sample transistor.
I have the following remarks:
- HEMT abbreviation is not explained in the paper (line 16).
- Figure 8 instead figure 9 should be in line 200.
- Figure 12 should be in line 239 instead of figure 9.
- In Table 4 should be example instead of exemple.
- Conclusions are too short and obvious. Please rewrite them.
Author Response
Response to Reviewer 2 Comments
The paper describes the thermal time constants of GaN HEMTs transistor. It also shows the model order reduction. Thermal issue are shown. It compares analythical model with numerical simulation. It would be better if the authors shown measurement results of a sample transistor.
Measurement Results (thermoreflectance measurements) have been provided and compared to the Finite Element Modeling
Point 1 : HEMT abbreviation is not explained in the paper (line 16).
Response 1 : HEMT has been defined on line 16
Point 2 : Figure 8 instead figure 9 should be in line 200.
Response 2 : Correction has been made
Pont 3 : Figure 12 should be in line 239 instead of figure 9.
Response 3 : Correction has been made
Point 4 : In Table 4 should be example instead of exemple.
Response 4 : Correction has been made
Point 5 : Conclusions are too short and obvious. Please rewrite them
Response 5 : new conclusion is proposed
In this work, we have addressed the understanding of the thermal time constants in GaN HEMTs. Its goal is not to propose the best physics-based approach to under-stand what happens in submicronic GaN HEMT device, but it is to propose an approach based on a well know theory (Fourier) and tools (Finite Element Method) that are commonly used in Labs and electronic industry.
In understanding, we mean that with the help of the article, the end user will grasp:
-that they are several thermal times constants in a HEMT, even in a 1D single lay-er.
-the influence of the GaN layer on the thermal times constants
-the evolution of these thermal time constants with the dimensions of the devices, and in particular with the gate length which is one of the key points to address high frequency applications.
The last conclusion to this work is that we have shown that with common tools and a Model Order Reduction technique, it is possible to extract real physical thermal time constants what is generally possible only using analytical approaches.
Reviewer 3 Report
The paper presents a comparison between numerical and analytical model to extract thermal parameters of GaN devices. It also presents the extraction of time constants and thermal resistances using model order reduction (MOR) technique to prove that this technique can merge both analytical and numerical modeling techniques. The paper is well organized and clear. However, the provided materials are light and not enough to be considered for journal publication. I rather suggest the authors to summarize the paper and resize it to be published as IEEE conference paper. Further, the following comments could be considered to improve the paper:
- Reference list is old (newest one is 2018 and 2016, and the remaining references are 10 years and older). Updated references are necessary to ensure the novelty of the proposed contribution.
- Figure 9 in line 200 should be Figure 10.
Author Response
Response to Reviewer 3 Comments
The paper presents a comparison between numerical and analytical model to extract thermal parameters of GaN devices. It also presents the extraction of time constants and thermal resistances using model order reduction (MOR) technique to prove that this technique can merge both analytical and numerical modeling techniques. The paper is well organized and clear. However, the provided materials are light and not enough to be considered for journal publication. I rather suggest the authors to summarize the paper and resize it to be published as IEEE conference paper. Further, the following comments could be considered to improve the paper:
Thank you for your comments. We have added some elements that increase the value of the paper
- Reference list is old (newest one is 2018 and 2016, and the remaining references are 10 years and older). Updated references are necessary to ensure the novelty of the proposed contribution.
Response 1 : Reference list has been updated with 5 new references. Model Order Reduction has been extensively studied in the past, but it is the first time in our knowledge that it is used it is shown that numerical extraction of thermal time constants with MOR provides the real thermal thermal time constants obtained by analytical solution of heat equation.
Point 2 : Figure 9 in line 200 should be Figure 10.
Response 2 : Correction has been made
Reviewer 4 Report
- The approach/model presented in 2.1 is too high-level. Please add a more explanatory formulation of the mentioned approach. Maybe a Figure/diagram would also help. My suggestion is to start from the basic heat equation and reach, by mathematical passages, your proposed formulation/modeling. Only referring to [7],[10] does not allow to follow the flow of this manuscript, while this paper should be somehow self-standing.
- Please better highlight the novelty with respect to [7],[10]. Also, please considering revising the title: the work seems to be an application/verification of a formulation in different test structures, rather than a tool for understanding.
- Please justify that "accuracy with fewer vectors is improved as compared to the use of eigenvectors" and that "the steady state temperature is always exact."
- Could you please better explain how to select the number 'm' of thermal time constants in the formulation to represent the time constant spectrum? Is that a standard process in [7], and how is it adapted in the MOR formulation?
- The authors are surely aware that thermal time constants are of similar order of magnitude with respect to charge-release time constants due to trapping at the interface layers (see: https://www.mdpi.com/2079-9292/10/2/137). How did you deal with trapping, are they not modeled in your simulations? What are the implications in practice? Please comment.
- The work would be more complete if the results could be somehow compared with experimental ones, even though found in references. At least, a comparative discussion with other works dealing with simulated or measurement thermal resistance/impedance in GaN would be necessary.
- The article is too colloquial in some parts and needs an English language review.
Author Response
Response to Reviewer 4 Comments
Point 1 : The approach/model presented in 2.1 is too high-level. Please add a more explanatory formulation of the mentioned approach. Maybe a Figure/diagram would also help. My suggestion is to start from the basic heat equation and reach, by mathematical passages, your proposed formulation/modeling. Only referring to [7],[10] does not allow to follow the flow of this manuscript, while this paper should be somehow self-standing.
Response 1 : A diagram has been added in the introduction to better understand the process.
Point 2 : Please better highlight the novelty with respect to [7],[10]. Also, please considering revising the title: the work seems to be an application/verification of a formulation in different test structures, rather than a tool for understanding.
Response 2 : In understanding, we mean that with the help of the article, the end user will grasp:
-that they are several thermal times constants in a HEMT, even in a 1D single layer.
-the influence of the GaN layer on the thermal times constants
-the evolution of these thermal time constants with the dimensions of the devices, and in particular with the gate length which is one of the keypoint to address high frequency applications.
The last conclusion of this work is that we have shown that with common tools and a Model Order Reduction technique, it is possible to extract real physical thermal time constants, what is generally possible only using analytical approaches.
Point 3 : Please justify that "accuracy with fewer vectors is improved as compared to the use of eigenvectors" and that "the steady state temperature is always exact."
Response 3 : When we use our Ritz vector approach. The first vector is obtained by solving the steady state system KT=F, the others are coming from orthogonal projection, so when t->infinity we are sure that the steady state is reached. With a pure eigenvalue decomposition, there are several major issues:
-the number n of numerical equations is very large, (few 100000) so it is very difficult to compute eigenvalues and eigenvectors
-the second difficulty is to select the main eigenvalues
-the steady state is not reached because
Point 4 : Could you please better explain how to select the number 'm' of thermal time constants in the formulation to represent the time constant spectrum? Is that a standard process in [7], and how is it adapted in the MOR formulation?
Response 4 : m depends on the precision you want on the reduced model. You can fix it as you want. Figure 4 show results for various m
In our article the extraction of the time constants is performed through model order reduction from the numerical system of Partial Differential Equation. What is extraordinary is that this model order reduction technique based finally on a numerical approach gives the real physical time constants. It is what it has been check with comparison with example that can be solved analytically.
The process to extract thermal time constants is very different in [7], because it is a fitting process associating with a modeling equation and a minimization of the error.
Point 5 : The authors are surely aware that thermal time constants are of similar order of magnitude with respect to charge-release time constants due to trapping at the interface layers (see: https://www.mdpi.com/2079-9292/10/2/137). How did you deal with trapping, are they not modeled in your simulations? What are the implications in practice? Please comment.
Response 5 : The simulation we present here is only a pure thermal simulation because only the heat equation is solved. Concerning the trapping effects, they are present when we perform measurements but it can be minimized with special operating conditions.
When the thermal part of the model is coupled to the electrical one, the model takes into account of the trapping process. We have published also several papers on trapping measurement and modeling.
Example :
K. Subramani, J. Couvidat, A. A. Hajjar, J. Nallatamby, R. Sommet and R. Quéré, "Identification of GaN Buffer Traps in Microwave Power AlGaN/GaN HEMTs Through Low Frequency S-Parameters Measurements and TCAD-Based Physical Device Simulations," in IEEE Journal of the Electron Devices Society, vol. 5, no. 3, pp. 175-181, May 2017, doi: 10.1109/JEDS.2017.2672685.
Round 2
Reviewer 1 Report
In my opinion the paper can be publish in its present version.
Author Response
Thank you for your comments
Reviewer 3 Report
I can not see the modifications and addressed comments in the new version. I don't have to read the whole paper again and compare with the old one to check if the comments are addressed. Please provide a better version with highlighted modifications and added materials.
Author Response
Thank you for your comments. We have added some elements that increase the value of the paper
- Reference list is old (newest one is 2018 and 2016, and the remaining references are 10 years and older). Updated references are necessary to ensure the novelty of the proposed contribution.
Response 1 : Reference list has been updated with 5 new references.
(added in the text)
The MOR has been extensively studied in the past years, however, to the best of our knowledge, this is the first time that it is used to demonstrate the extraction of thermal time constants with MOR, that provides the real thermal time constants obtained by analytical solution of the heat equation.
Point 2 : Figure 9 in line 200 should be Figure 10.
Response 2 : Correction has been made
please provide a better version with highlighted modifications and added materials.
Response : changes added to the original paper are now in red. It contains many improvements
Reviewer 4 Report
Response to point 2, the difference with previously published work is not well addressed. Please include the novel aspects here with respect to [7] and [10] and discuss them in the article.
Response to point 3 seems not to be complete!
For all the points: it seems that the answers provided have not been used to improve the manuscript. Also, I cannot find a version with highlighted changes.
Author Response
Point 1 : The approach/model presented in 2.1 is too high-level. Please add a more explanatory formulation of the mentioned approach. Maybe a Figure/diagram would also help. My suggestion is to start from the basic heat equation and reach, by mathematical passages, your proposed formulation/modeling. Only referring to [7],[10] does not allow to follow the flow of this manuscript, while this paper should be somehow self-standing.
Response 1 : Figure 1 presents a flow chart in the introduction to better understand the process from heat equation to the reduced model.
Details are also added in the text :
The thermal model is obtained from the Finite Element formulation of the classical heat equation based on the Fourier phenomenological approach. The assumptions are adiabatic conditions on lateral directions, uniform heat flux on the surface that represents the gate of the device, and a fixed baseplate temperature.
Response to point 2, the difference with previously published work is not well addressed. Please include the novel aspects here with respect to [7] and [10] and discuss them in the article.
Response to point 2 : (added in the paper)
Now Ref. 10 is Ref. 12
In this paper, a comparison between the thermal time constants and thermal resistances obtained both using the analytical model [7] and numerical model (ANSYS software) coupled with MOR [12] has been carried out. This work can be considered as an extended study of [7]. However, in our article the time constants are extracted through model order reduction from a numerical Partial Differential Equation. The excellent part is that this model order reduction technique is based on a numerical approach, thereby yielding the real physical time constants. The way of extracting the thermal time constants is found to be quite different in Ref. [7], as because the authors used a fitting procedure associating with a modeling equation and a minimization of an error function.
The motive of this work is essentially to prove that the real physical time constants can be determined by using our MOR technique.
Response to point 3 seems not to be complete!
Point 3 : Please justify that "accuracy with fewer vectors is improved as compared to the use of eigenvectors" and that "the steady state temperature is always exact." (added in the paper)
The main advantages of the Ritz vector method are that important response modes are not neglected and the accuracy with fewer vectors is improved compared to the use of eigenvectors. Moreover, using this method, the steady state temperature is always exacted. Indeed, the first Ritz vector is obtained by solving the steady state KT=F. The other vectors originate from orthogonal projections. When the time is infinite, their contribution decreases and tends to zero. With a pure eigenvalue decomposition, there are several major issues:
-the number n of numerical equations is very large, (few 100000) so it is very difficult to compute all the eigenvalues and the eigenvectors
-the second difficulty involves in selecting the main eigenvalues
-the steady state is not reached if all the eigenvalues are not considered
Point 4 : Could you please better explain how to select the number 'm' of thermal time constants in the formulation to represent the time constant spectrum? Is that a standard process in [7], and how is it adapted in the MOR formulation?
(Figure5 shows the influence of m and shows also that the steady state is always reached).
(added in the paper)
Figure 5 shows the influence of the number of Ritz value. In the first example, we can observe that with 5 values, a very good accuracy is obtained from 1ns to the steady state, while this one is reached in all cases.
The way of extracting the thermal time constants is found to be quite different in Ref. [7], as because the authors used a fitting procedure associating with a modeling equation and a minimization of an error function.
For all the points: it seems that the answers provided have not been used to improve the manuscript. Also, I cannot find a version with highlighted changes.
Response : The answers have been added to this version of the paper in red. Moreover many improvements have been added to the original paper.The conclusion has been expanded to clarify the message the authors wish to deliver in this paper.
Round 3
Reviewer 3 Report
The authors properly addressed the raised comments. The manuscript could be published in present form.
Thank you.
Reviewer 4 Report
The authors have now answered the questions and corresponingly applied the changes to the manuscript.
