Ken Haruta, Shiuh-Wuu Lee, Colin McAndrew, Bernd Meinerzhagen, Laurence W. Nagel, E. James Prendergast
©SHUTTERSTOCK.COM/IAROSLAV NELIUBOV
Hermann K. Gummel made numerous fundamental and high-impact contributions to the IC industry, in particular to IC design technologies. One of the most notable was in the domain of bipolar transistors: Today nobody designs bipolar transistors, or circuits based on bipolar transistors, without using the eponymous “Gummel plot.” An early pioneer of the IC industry, Hermann passed away in 2022, the year of the 75th anniversary of the invention of the transistor. Here, six of his former coworkers review some of Hermann’s most significant and wide-reaching contributions, to bipolar transistor modeling in particular and to computer-aided IC design technology in general, and reminisce about his warm and exceptional personality.
Today the design and progression of ICs is only possible because of the development of a large number of computer-based tools. The steady exponential growth of the IC industry over many decades following Moore’s law would have been impossible without the extensive use of CAD technology. The progress of this design technology has been driven by the expanding scope and requirements of IC design, which started shortly after Robert Noyce invented the first silicon IC in 1959. The most important innovator in the early days of CAD technology for ICs is, without any doubt, Hermann K. Gummel, who passed away in 2022, the year of the 75th birthday of the bipolar transistor, which is particularly connected to his name. Today, nobody designs bipolar transistors without using a Gummel plot.
Therefore, it is the right time to honor Hermann K. Gummel’s fundamental contributions in several key domains of CAD technology. In this article six former coworkers of Hermann have joined forces to provide an overview of Hermann’s work and their personal views on his outstanding personality. The article is organized in four sections written by the different authors, highlighting different fields of CAD technology that have benefited most from Hermann’s significant contributions.
The history of numerical modeling for semiconductor devices starts with Hermann’s landmark paper from 1964 [1], bearing the modest title “A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations.” In this paper of 10 pages, which appeared in the October issue of IEEE Transactions on Electron Devices, he presented the nonlinear solution method that is still one of the most important algorithms within modern technology CAD (TCAD) models. Today it is typically addressed as Gummel’s nonlinear relaxation method or just Gummel’s method. One has to remember that, prior to 1964, computers had “core” memory sizes of a few thousand bytes and clock cycle times below 1 MHz, and would crash daily. Therefore, only algorithms that needed little memory on one hand and had a robust and fast convergence behavior on the other could be used. This is exactly what Hermann’s new algorithm offered for most applications. This algorithmic breakthrough, and the availability of accurate numerical solutions, created a boost for the art of bipolar transistor modeling in particular and for the numerical modeling of semiconductor devices in general, and it paved for the way to many important innovations in the years to come. Besides this key solution algorithm, Hermann’s paper from 1964 provides a wealth of additional information as most of his papers do. The most important details about the proper modeling of mobilities and recombination processes for silicon are included, and the typical alternatives for the formulation of boundary conditions are discussed as well. The possibility of computing small-signal parameters in a numerically stable manner is mentioned, and a method for including parasitic circuit elements is addressed. Moreover, a possible extension of the 1D model in [1], allowing the efficient modeling of the base crowding effect by coupling several 1D bipolar models in the base, is outlined. This idea was later realized by other groups (see, for example, [5]) and proved very useful for the further scaling of bipolar transistors. Finally, an efficient and simple method for modeling the self-heating of semiconductor devices, which is still used today, is described in [1].
The availability of the 1D numerical model allowed Hermann and his coworkers new insights into bipolar effects like high injection and base push-out [3], which were controversial issues at that time. Moreover, it allowed the investigation of complicated device operation like a silicon diode, backward biased in a strong impact ionization mode (a Read diode) and placed in a small resonant circuit environment allowing the generation of autonomous microwave oscillations. Hermann and his coworker Donald Scharfetter studied this device for several years as some lesser known publications show (e.g., [4]). Finally, they summarized their results in Hermann’s second landmark paper, “Large Signal Analysis of a Silicon Read Diode Oscillator,” which appeared in the January 1969 issue of IEEE Transactions on Electron Devices [2] and described the first transient numerical device simulation. This new method allowed the computation of the limit cycle of a microwave oscillator, exploiting the strong nonlinearity of impact ionization in silicon. The transient analysis is based on a semi-implicit time integration method that enhances the stability of time integration, but the key detail mentioned mostly in the context of this article is the Scharfetter–Gummel discretization method for the particle current densities between two grid nodes. This discretization formula is explicitly given in this article for the first time but probably already existed in a hidden manner in [1], where the same principles are used for the discretization of electron and hole particle current densities. This discretization method and/or generalizations of it are adopted in virtually every TCAD simulator in use today since it turned out to be very difficult to generate stable TCAD solutions on coarse grids without this method.
Hermann’s early paper [1] indicates that he was one of the most sophisticated scientific programmers of his time, and, amazingly, he managed to keep this skill level through the demanding times as lab director and well beyond his official retirement. Being on the forefront of the development of CAD for ICs for decades made him an ideal mentor for the younger engineers and scientists at Bell Labs and beyond. Discussing a technical problem with him was a career highlight and an unforgettable moment for everybody who was able to enjoy his deep and broad knowledge. Generations of young engineers and scientists considered it an honor to present their work when he was present. His feedback was typically very polite but firm when he detected weaknesses. One example is his doubt about the general validity of the Ward–Dutton charge-oriented model [6] for the low-frequency terminal current modeling of MOSFETs. His doubt triggered a fundamental research project, which he closely monitored, that is summarized in the Habilitation-Thesis of Heinz K. Dirks [7]. The answer is that the ansatz in [6] is a good approximation for symmetric MOSFETs but fails for nonsymmetric transistors. Hermann actively supported the CAD activities at many universities and was personally involved in making them successful.
Hermann’s language was always very precise, and he disliked overstatements. For example, he always used the term Chynoweth’s formula and refused to use the term Chynoweth’s law as most people do for the formula that describes impact ionization in most TCAD simulations. Even decades after his official retirement, Hermann seemed to always be online. E-mails were answered in perfect English within hours, and sometimes even minutes, and we could count on his support when we needed it.
Bernd Meinerzhagen and E. James Prendergast
In 1970, Hermann published a very efficient method for the calculation of multiterminal resistances and capacitances with irregular shapes in two dimensions [8]. The method is based on Cauchy’s integral formula, which states that “a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk.” Hermann made ingenious use of this and developed a software tool called RESCAL. Its original purpose was to calculate resistances, but it turned out to be very useful as well for calculating capacitances in IC designs. However, in the original version of RESCAL there were some limitations on the shape of the boundary. Consequently, when he retired as lab director and continued working as a consultant for Bell Labs, Hermann had more time and wanted to expand the capabilities of RESCAL to more general cases. For this project, he was looking for someone to work with him, and one day a department head in Hermann’s former lab asked me if I would like to work with him. I told him that I had heard Hermann was a workaholic and a “slave driver,” so I was a bit hesitant. Then he suggested I have a one-on-one meeting to talk with Hermann. At the meeting, I asked Hermann bluntly about his reputation. He told me it was far from the truth. He would welcome people to work extra hours if they were interested enough in their work, but he had never required or even requested them to work overtime. Even when another director asked him to get his people to do extra work because that director’s group was behind schedule, Hermann told me he had flatly refused because it was their problem, and he was not going to ask his own people to make up for it. So, I decided to take up the offer to work with him on RESCAL. It became a very useful tool, and I applied it to numerous cases in the following years (see Figure 1 for a typical RESCAL simulation result). It turned out that the time I worked with Hermann was the best period in my career at Bell Labs. Hermann was a genius—well, a supergenius—but he was also a warm, kind gentleman. (Figure 2 shows Hermann at a meeting with some of the authors.)
Figure 1. Numerical solution of 2D Laplace’s equation within a star-shaped domain calculated by RESCAL. Constant Dirichlet boundary conditions are applied on the thick boundary lines, and homogeneous Neumann boundary conditions on the other parts of the boundary. Inside the star, dashed lines are field lines, and solid lines are equipotential lines.
Figure 2. Picture from the state of the lab meeting in 1987 with Hermann K. Gummel (second row, second from left), Ken Haruta (first row, far right), Bernd Meinerzhagen (third row, far left), Laurence W. Nagel (third row, second from right), and E. James Prendergast (second row, far left).
Ken Haruta
Hermann’s contributions to, and influence over, compact models (i.e., models that are used for circuit simulation for IC design) and how compact models should be formulated, tested, and verified loom large to this day.
Hermann’s first foray into compact modeling was in 1969 [9], on the depletion capacitance of pn-junction diodes. Standard theory predicts that, for a voltage V applied across a junction, the capacitance is ${C} = {C}_{0} / {(}{1}{-}{V} / {\phi}_{\text{bi}}{)}^{m}$, where ${C}_{0}$ is the capacitance at ${V} = {0}$, ${\phi}_{\text{bi}}$ is the built-in potential of the junction, and m is a coefficient of around 1/3 to 1/2. This form has an obvious numerical problem as V approaches ${\phi}_{\text{bi}}$ in forward bias: it blows up to infinity! This issue could be avoided when analysis was done by hand, but it was a big problem as circuit analysis programs became more widely used in the late 1960s and early 1970s. The first proposed solution to this problem was from Hermann [9]. Moreover, his approach gives physically correct behavior: C does not “blow up” but peaks as V approaches ${\phi}_{\text{bi}}$, and then decreases to zero for ${V}\,{≫}\,{\phi}_{\text{bi}}$, exactly as it should because the depletion charge (and hence the capacitance) disappears. Many modern formulations for depletion charge modeling are derivatives of this approach.
Hermann’s most far-reaching contributions in compact modeling, from 1970, were the formulation of the integral charge-control relation (ICCR) [10] and the application of that to a compact model for bipolar junction transistors (BJTs) in [11], which gives what is now called the Gummel–Poon, or GP, model. Shockley had developed the ideal diode model in the 1940s, which shows that the diode current depends exponentially on V, based on pioneering physical analysis detailed in his book [12]. Subsequent BJT models were predicated on that, and despite invoking some “physical” analyses, were based primarily on empirical ${\alpha} = {I}_{C} / {I}_{E}$ and ${\beta} = {I}_{C} / {I}_{B}$ parameters, where ${I}_{C}$, ${I}_{E}$, and ${I}_{B}$ are the collector, emitter, and base currents, respectively. As the use of BJTs expanded, and a more accurate and detailed understanding as well as the modeling of nonideal effects were needed, it was essentially impossible to “shoehorn” into these empirically based BJT models things like the Early effect (modulation of ${I}_{C}$ by the collector voltage ${V}_{C})$ and high-level injection (decrease in ${\beta}$ at a high base voltage ${V}_{B}{)}$.
Hermann’s ICCR and the GP model physically, naturally, and elegantly embody the fundamentals of how BJTs operate. They represent a revolution in the understanding of BJT behavior and of how a good compact model should be developed: based on physics, but with insight and intuition as to how to derive relatively simple, yet accurate, algebraic equations from complex partial differential equations. The GP model gives the collector–emitter (i.e., transport) current as \begin{align*}{I}_{\text{CE}} = & {A}_{E} \frac{{q}\,{\bullet}\,{n}_{i}^{2}\,{\bullet}\,{\phi}_{t}\,{\bullet}\,{\mu}}{\mathop{\int}\nolimits_{0}\nolimits^{{t}_{B}}{{N}_{B}(x)dx}} \\ & {\times}\,\frac{{\exp}{\left(\frac{{V}_{B}}{{\phi}_{t}}\right)}{-}{\exp}{\left(\frac{{V}_{C}}{{\phi}_{t}}\right)}}{{q}_{B}} \end{align*} where ${A}_{E}$ is the area of the emitter, q is the elementary charge, ${n}_{i}$ is the intrinsic carrier concentration, ${\phi}_{t}$ is the thermal voltage, ${\mu}$ is mobility, ${t}_{B}$ is the base thickness, ${N}_{B}{(x)}$ is the doping level in the base (which depends on position x in the base, from the emitter to the collector), and ${q}_{B}$ is the so-called normalized based charge (which has a value of one with no bias applied). The Early effect and modeling of high-level injection appear naturally in Hermann’s formulation through the bias dependence of ${q}_{B}$ (see [11] for details).
One of the most beautiful features of this model is that it captures, to first order, all important aspects of BJT behavior: the dependence on geometry (via ${A}_{E})$; the dependence on temperature (directly through ${\phi}_{t}$ and indirectly though ${n}_{i}$, which depends strongly on temperature, and ${\mu}$, which depends weakly on temperature); the dependence on the structural (and designable) parameters ${t}_{B}$ and ${N}_{B}$; the dependence on bias (directly, and through ${q}_{B})$; and, indirectly, statistical variations (through variations in ${A}_{E}$, ${t}_{B}$, and${N}_{B}$).
Hermann’s ICCR and the GP model are works of ingenuity and formed the basis of all of the more advanced BJT compact models developed since the ICCR and GP model were introduced, way back in 1970. In our opinion, they also form by far the best “mental” picture of how a BJT works; ${\alpha}$ and ${\beta}$ should have vanished into the mists of time but, alas, have not done so yet (more than 50 years since Hermann blazed the path to the future!).
Given the importance of the dopant sheet density in the base, ${\int}_{0}^{{t}_{B}}{N}_{B}{(x)}{dx}$ in the previous expression, this quantity is now referred to as the Gummel number (that name also covers variants that may include ${\phi}_{t}\,{\bullet}\,{\mu}$ and/or ${n}_{i}$, but it is the base thickness and doping profile that are key). Verification of a BJT model versus data is now universally done via a log-linear plot of ${I}_{C}$ and ${I}_{B}$ versus ${V}_{B}$, which is called the Gummel plot [13].
As CMOS technology became dominant, the need for accurate MOS transistor models escalated. Multiple approaches were investigated, and many of these worked well enough for digital applications but had significant shortcomings for analog and RF applications. Complaints about, and benchmarks to test, capabilities of MOS transistor models from the perspective of circuit designers were voiced in [14] and [15]. However, it was not obvious how to translate some design requirements into “physical” benchmarks that model developers could understand.
The common perception that the MOS transistor drain current ${I}_{D}$ varies exponentially with gate bias ${V}_{G}$ in weak inversion (“subthreshold”) is incorrect—the deviation from exact exponential variation is critical for the design of modern low-power circuits, yet “threshold voltage”-based MOS transistor models completely missed this. How do you test if a model captures this physical behavior?
In strong inversion (high ${V}_{G})$, at low drain bias, ${I}_{D}$ varies approximately linearly with drain voltage ${V}_{D}$. In weak inversion (low ${V}_{G}$), ${I}_{D}$ varies as ${1}{-}{\exp}{(}{-}{V}_{D} / {\phi}_{t}{)}$. How do you test if a model correctly captures this behavior and transitions smoothly from low to high ${V}_{G}$?
The source and drain of most MOS transistors are symmetric. How do you determine if a MOS transistor model (which has four terminals) is properly symmetric? (This is critical for modeling the behavior of some specific RF CMOS circuits.)
Hermann’s knowledge of devices and models led him to develop benchmark tests for each of these three behaviors (see [16], where the tests developed by Hermann were first presented outside Bell Labs). They are known as the Gummel tree-top test, the Gummel slope ratio test, and the Gummel symmetry test, respectively. All modern MOS transistor models are now required to pass those tests.
There are other hidden gems in Hermann’s work that are often overlooked (which is why it is important to read everything in all of his papers carefully).
First, many people have “discovered” that a plot of ${\log}{(}{I}_{C}{)}$ versus ${V}_{B}$ is not exactly a straight line with slope ${1} / {\phi}_{t}$, as the preceding GP equation looks like it predicts. They therefore “improve” the model by replacing ${\phi}_{t}$ with ${n}\,{\bullet}\,{\phi}_{t}$, where n is some “ideality” coefficient that can be different from 1. There are reasons to include such a coefficient, especially for heterojunction bipolar transistors because some (second-order) physics is omitted in the derivation of the ICCR and the GP model. However, by far the most common reasons for deviation from the simple theory predictions are that the measurement temperature is not controlled accurately, and that the reverse Early effect (“emitter capacitance”) is not taken into account (which it is in the GP model). Both of these are explicitly pointed out in [11] but are still widely overlooked.
Second, the “classic” view of ${f}_{T}$ from the 1950s, yet still in widespread use today, is that it is “the frequency where the small signal version of ${\beta} = {1}$.” This definition has many shortcomings. Hermann’s little-known article [17] is the key bridge between the classic and modern views of what ${f}_{T}$ is.
Shiuh-Wuu Lee and Colin McAndrew
I first met Hermann in 1973 when I was interviewing for a job at Bell Laboratories. At that time, Hermann already had pioneered the use of what we now call TCAD tools to understand the operation of the BJT, he had derived the integral charge relation for the BJT, and with Sam Poon he had published the Gummel–Poon model for circuit simulators. In fact, I had implemented the Gummel–Poon model in Spice at that point, and I had developed an enormous respect for the elegance of the integral charge relation. When I interviewed Hermann, I was amazed at how quiet and shy he was, and I found myself wondering how somebody so brilliant and so successful could be so humble.
For most engineers, Hermann’s work with BJTs would be sufficient for a distinguished career with the accolades that accompany such accomplishments. But Hermann was just getting started. At this point, Hermann was a second-level manager responsible for a department of 30 engineers, but he still made time for engineering breakthroughs. Hermann’s department was part of a laboratory that was using polycells and CAD tools to develop custom MOS ICs for the Bell System. While polycells were very successful in rapid prototyping of custom ICs, our department was facing some real design issues. The circuits that were developed were too large to be simulated by ADVICE, the Bell Labs version of Spice that I had developed. But the Bell Labs logic simulator, called LAMP, which could easily accommodate the size of the circuits, was not accurate enough to detect the timing errors that were plaguing the MOS circuits.
In the early 1970s, Hermann conceived the idea of a timing simulator that would run much faster and accommodate much larger circuits, compared to ADVICE, yet provide sufficient accuracy for MOS circuits. I remember Hermann calling me at the IEEE International Solid-State Circuits Conference to tell me that he was thrilled that he couldn’t sleep that night because he came up with an idea for a faster simulator and he wanted to run it past me. I was amazed at the idea and enthusiastically encouraged Hermann to pursue it.
Hermann’s work resulted in the prototype program MOTIS (MOS Timing Simulator) [18], which was the beginning of “fast Spice” timing simulators—more accurate than logic simulators and much faster than circuit simulators. Hermann combined the computational techniques of table lookup models, implicit integration, and a direct solution technique to accomplish this task. To quote A. Richard Newton, Dean of the UC Berkeley College of Engineering, “While I think Hermann would agree with me that none of the specific mathematical techniques used in MOTIS was new in and of itself, it was the combination of techniques, data structures, and code that produced a simulation system that had a very broad impact. Once the work was published, it was implemented and extended in virtually every major semiconductor house and at many universities throughout the world. I was personally involved in the development of one such version, MOTIS-C at Berkeley, and so I can personally attest to both the elegance and the engineering insight Hermann gave us all in that work.” [19].
Hermann had an uncanny intuition about the design problems that the ever-increasing size of ICs—fueled by Moore’s law—were impressing on CAD tools. Hermann realized long before many experts in the IC business that computers were the only way to address complex circuit design problems, just as he had realized that computers were the only way to address semiconductor physics problems 20 years earlier.
One example of a CAD tool that Hermann pioneered was his SCHEMA program, which allowed a designer to enter a circuit description graphically by entering a schematic of the circuit. At the time SCHEMA was developed, the traditional manner of entering a circuit description was a textual “netlist” file that was easily understood by a simulator but very error prone and painful for a designer. Hermann would sit for hours with designers observing how they entered netlists for simulation, which led him to understand that a schematic entry tool would save designers an enormous amount of time.
Hermann also built a very successful graphical editor, called GRED, and this also illustrates how he first came to understand the human interface before developing a CAD tool. I personally witnessed Hermann sitting beside designers for hours on end, watching how they did their work, how they implemented commands, and observing the number of keystrokes needed, all the while taking notes and thinking about ways to improve their efficiency. He then developed GRED from the notes he had gathered.
Hermann was also the first to build and publish a practical device extraction tool. His HCAP program [20] allowed a designer to extract a netlist from the physical layout of an IC. Before HCAP, designers used colored pencils and a Calcomp plot of the layout to check that the layout was indeed the circuit the designer intended.
Hermann was a true visionary and had a clear understanding that the size and complexity of ICs was increasing much faster than a human’s ability to cope. CAD tools were now a necessary part of the IC design process, and Hermann was at the forefront of electronic design automation tool development, while managing a department that had now grown to more than 100 engineers at two Bell Labs locations.
Laurence W. Nagel
When Hermann K. Gummel passed away, we lost one of the key founding fathers of IC industry. He was an outstanding scientist, a great lab manager and leader, and a most admired individual and mentor, all at the same time. He led by example, an example we all had difficulty keeping up with, but one that certainly shaped our own approach toward science and management. He remains the inspiration for all of us.
Ken Haruta (haruta@alum.mit.edu) , retired, was with Bell Labs, Allentown, PA 18103 USA.
Shiuh-Wuu Lee (shiuhwuu.lee@gmail.com) resides in Saratoga, California, US 95070 USA.
Colin McAndrew (mcandrew@ieee.org) is with NXP Semiconductors, Chandler, AZ 85224 USA.
Bernd Meinerzhagen (b.meinerzhagen@tu-bs.de) is with Technische Universität Braunschweig, Fakultät für Elektrotechnik, Informationstechnik, Physik, 38106 Braunschweig, Germany.
Laurence W. Nagel (lwn@omega-enterprises.net) is with Omega Enterprises Consulting, Kensington, CA 94708 USA.
E. James Prendergast (jim@prenkin.com) resides in Durango, CO 81301 USA.
[1] H. K. Gummel, “A self-consistent iterative scheme for one-dimensional steady state transistor calculations,” IEEE Trans. Electron Devices, vol. ED-11, no. 10, pp. 455–465, Oct. 1964, doi: 10.1109/T-ED.1964.15364.
[2] D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator,” IEEE Trans. Electron Devices, vol. ED-16, no. 1, pp. 64–77, Jan. 1969, doi: 10.1109/T-ED.1969.16566.
[3] H. C. Poon, H. K. Gummel, and D. L. Scharfetter, “High-injection in a bipolar transistor,” IEEE Trans. Electron Devices, vol. ED-16, no. 5, pp. 455–457, May 1969, doi: 10.1109/T-ED.1969.16777.
[4] H. K. Gummel and D. L. Scharfetter, “Avalanche region of IMPATT diodes,” Bell Syst. Tech. J., vol. 45, no. 10, pp. 1797–1827, Dec. 1966, doi: 10.1002/j.1538-7305.1966.tb02436.x.
[5] W. L. Engl and H. K. Dirks, “Functional device simulation by merging numerical building blocks,” in Numerical Analysis of Semiconductor Devices and Integrated Circuits, B. T. Brown and J. J. H. Miller Eds. Dublin, Ireland: Boole Press, 1981, pp. 34–62.
[6] D. E. Ward and R. W. Dutton, “A charge-oriented model for MOS-transistor capacitances,” IEEE J. Solid-State Circuits, vol. SSC-13, no. 5, pp. 703–708, Oct. 1978, doi: 10.1109/JSSC.1978.1051123.
[7] H. K. Dirks, “Kapazitätskoeffizienten nichtlinearer dissipativer systeme,” Habilitation thesis, RWTH Aachen University, Aachen, Germany, 1988.
[8] B. R. Chawla and H. K. Gummel, “A boundary technique for calculation of distributed resistance,” IEEE Trans. Electron Devices, vol. ED-17, no. 10, pp. 915–925, Oct. 1970, doi: 10.1109/T-ED.1970.17095.
[9] H. C. Poon and H. K. Gummel, “Modeling of emitter capacitance,” Proc. IEEE, vol. 57, no. 12, pp. 2181–2182, Dec. 1969, doi: 10.1109/PROC.1969.7529.
[10] H. K. Gummel, “A charge control relation for bipolar transistors,” Bell Syst. Tech. J., vol. 49, no. 1, pp. 115–120, May 1970, doi: 10.1002/j.1538-7305.1970.tb01759.x.
[11] H. K. Gummel and H. C. Poon, “An integral charge control model of bipolar transistors,” Bell Syst. Tech. J., vol. 49, no. 5, pp. 827–852, May 1970, doi: 10.1002/j.1538-7305.1970.tb01803.x.
[12] W. Shockley, Electrons and Holes in Semiconductors. Princeton, NJ, USA: Van Nostrand, 1950.
[13] “Gummel plot.” Wikipedia. Accessed: Feb. 1, 2023. [Online] . Available: https://en.wikipedia.org/wiki/Gummel_plot
[14] Y. Tsividis, “Problems with precision modeling of analog MOS LSI,” in Proc. Int. Electron Devices Meeting, 1982, pp. 274–277, doi: 10.1109/IEDM.1982.190272.
[15] Y. Tsividis and K. Suyama, “MOSFET modeling for analog circuit CAD: Problems and prospects,” IEEE J. Solid-State Circuits, vol. 29, no. 3, pp. 210–216, Mar. 1994, doi: 10.1109/4.278342.
[16] C. McAndrew, H. K. Gummel, and K. Singhal, “Benchmarks for compact MOSFET models,” in Proc. Sematech Workshop Compact Models, 1995.
[17] H. K. Gummel, “On the definition of the cutoff frequency fT,” Proc. IEEE, vol. 57, no. 12, pp. 2159–2159, Dec. 1969, doi: 10.1109/PROC.1969.7509.
[18] B. R. Chawla, H. K. Gummel, and P. Kozak, “MOTIS – An MOS timing simulator,” IEEE Trans. Circuits Syst., vol. CAS-22, no. 12, pp. 901–910, Dec. 1975, doi: 10.1109/TCS.1975.1084003.
[19] A. R. Newton, Presentation of the 1994 Phil Kaufman Award to Dr. Hermann K. Gummel, Nov. 1994. [Online] . Available: https://people.eecs.berkeley.edu/∼newton/Presentations/Kaufman/HKGPresent.html
[20] P. A. Swartz, B. R. Chawla, T. R. Luczejko, K. Mednick, and H. K. Gummel, “HCAP – A topological analysis program for IC mask artwork,” in Proc. IEEE Int. Conf. Comput. Des., Port Chester, NY, USA, Oct./Nov. 1983, pp. 298–301.
Digital Object Identifier 10.1109/MED.2023.3262750