Ramses van der Toorn | Delft University of Technology (original) (raw)

Papers by Ramses van der Toorn

We present two complementary, analytical calculations of the electric field, stored charge, capac... more We present two complementary, analytical calculations of the electric field, stored charge, capacitance and transit time of the collector of a III-V HBT, for respectively low- and high bias regimes. Voltage, current and temperature dependence of base-collector capacitance as well as f/sub T/-peaking at high bias is thus quantitatively related to GaAs-like electron drift-velocity characteristics. The physical insight developed serves as a basis for extension of the Mextram compact bipolar transistor model for circuit simulation.

Journal of Marine Science: Research & Development, Jun 27, 2015

Axioms, Dec 27, 2022

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

We extend the category of parameter extraction methods that utilize weak avalanche under fixed em... more We extend the category of parameter extraction methods that utilize weak avalanche under fixed emitter current conditions to extract base resistance. Our new method consistently accounts for self-heating and Early effect so as to arrive at an accurate extraction of both base and thermal resistance. It involves only small variations in bias and temperature conditions, so that parameter extraction as a function of bias and temperature is enabled. The method is directly applicable to transistors - no dedicated test structures are needed - so that the method is cost effective. The method is demonstrated on RF SiGe HBTs.

We enhance Frobenius' method for solving linear ordinary differential equations about regular sin... more We enhance Frobenius' method for solving linear ordinary differential equations about regular singular points. Key to Frobenius' approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel's equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order.

Legendre’s equation is key in various branches of physics. Its general solution is a linear funct... more Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.

Abstract—As the demands for high operating frequency and high output power of modern bipolar tran... more Abstract—As the demands for high operating frequency and high output power of modern bipolar transistor circuits increase, designers are trying to exploit transistor operating regions where they would be able satisfy both conditions, namely working within an avalanche region. For such applications, compact models have to be at their best, so the designers could have a full overview of the device behaviour and possibilities. In order to accomplish that, there should be an adequate way in characterization of avalanche in the RF regime, as well as in verification of the devices against some figures of merit. This paper gives an insight on possible RF avalanche characterization techniques. Repercussions of working in the avalanche regime on some important transistor properties like unilateral power gain, the maximum avaliable gain and the stability factor is demonstrated. Measurements of modern industrial devices, as well as simulations using the world standard bipolar compact model Mex...

Mextram 504 is the new version of the Philips compact model for bipolar transistors. This documen... more Mextram 504 is the new version of the Philips compact model for bipolar transistors. This document contains the derivation of all the equations that are part of the model. This includes the description of the equivalent circuit of Mextram, the equations for the currents and the charges, the temperature-scaling model, the way self-heating is handled, and the noise model. We then discuss some small-signal approximations and the basis of geometric scaling. The formal model definition needed for implementation is given in another report NL-UR 2000/811 [ 1], parameter extraction is discussed in the report NL-UR 2001/801 [ 2]. These reports and the source code can be found on our web-site [ 3]. c ©Koninklijke Philips Electronics N.V. 2002 iii NL-UR 2002/806— March 2002 Model derivation of Mextram 504 Unclassified report Preface The Mextram bipolar transistor model has been put in the public domain in Januari 1994. At that time level 503.1 of Mextram was used within Koninklijke Philips Ele...

Absfroct- We pment two complementary, analytical calculatlons of the electric field, stored charg... more Absfroct- We pment two complementary, analytical calculatlons of the electric field, stored charge, capacitance and transit time of the collector of a IU-V HBT, for respectively low- and high bias regimes. Voltage, current and temperature dependence of basesollector capacitance as well as Jr-peaking at high hips is thus quantitatively related to GaAs-like electron drift-velocity E characteristics. The physical insight developed will serve as a basis for extension of the Mextram compact bipolar transistor model for circuit simulation. j -- 'g a 10

In this paper an extensive verification of the extraction method (published earlier) that consist... more In this paper an extensive verification of the extraction method (published earlier) that consistently accounts for self-heating and Early effect to accurately extract both base and thermal resistance of bipolar junction transistors is presented. The method verification is demonstrated on advanced RF SiGe HBTs were the extracted results for the thermal resistance are compared with those from another published method that ignores the effect of Early effect on internal base-emitter voltage and the extracted results of the base resistance are compared with those determined from noise measurements. A self-consistency of our method in the extracted base resistance and thermal resistance using compact model simulation results is also carried out in order to study the level of accuracy of the method. Keywords—Avalanche, Base resistance, Bipolar transistor, Compact modeling, Early voltage, Thermal resistance, Self-heating, parameter extraction.

This document presents the definition of the CMC world stardard model Mextram, for vertical bipol... more This document presents the definition of the CMC world stardard model Mextram, for vertical bipolar transistors. The goal of this document is to present the full definition of the model, including the parameter set, the equivalent circuit and all the equations for currents, charges and noise sources. Apart from the definition also an introduction into the physical background is given. We have given also a very basic parameter extraction procedure. Both the background and the parameter extraction are documented separately in dedicated documents. The transition from Mextram 503 to Mextram 504 is described, to enable the translation of a 503 parameter-set to a 504 parameterset. At last we have given some numerical examples that can act as a test of implementation. Delft University of Technology iii June 16, 2009 Mextram version 504.8 Mextram definition document Preface October 2004 The Mextram bipolar transistor model has been put in the public domain in Januari 1994. At that time leve...

AIMS Mathematics

Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the und... more Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the underlying spherical symmetry van der Toorn, Ramses Abstract: We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly, why the phase velocity of these wave modes is insensitive to their orientation with respect to the axis of rotation of the planet, while at the same time the very rotation of the planet is a precondition for the existence of the waves. As we show, answers to both questions can be rooted in Lie group and representation theory. In our study the Rossby-Haurwitz modes emerge in a coordinate-free, as well as in a Ricci tensor rank-free manner. We find them with respect to a continuum of spherical coordinate systems, that are arbitrarily oriented with respect to the planet. Furthermore, we show that, in the same sense in which the Lie derivative of Ricci tensor fields is rankfree, the wave equation for Rossby-Haurwitz modes is rank-free. We find that, for each irreducible representation of so(3), there is a corresponding sufficient condition for existence of Rossby-Haurwitz modes as solutions that are separable with respect to space and time. This condition comes in the form of a system of equations of motion for the coordinate systems. Coordinate systems that move along with Rossby-Haurwitz modes emerge as special cases of these. In these coordinate systems the waves appear as stationary spatial fields, so that the motion of the coordinate system coincides with the wave phase propagation. The general solution of the existence condition is a continuum of moving spherical coordinate systems that precess about the axes of the Rossby-Haurwitz modes. Within a single irreducible representation of so(3), the waves are dispersionless.

We adapt the Mextram 504 compact model for Si and SiGe bipolar transistors to make it suitable fo... more We adapt the Mextram 504 compact model for Si and SiGe bipolar transistors to make it suitable for compact modeling of GaAs HBT's. We discuss the different physics included in the new model, Mextram 3500, and demonstrate the capabilities of the new model on GaAs HBT characteristic simulations. We also show an example of advanced GaAs PA-circuit simulations that have

Gallium Arsenide Integrated Circuit Annual IEEE Symposium, 2003

We present a physically based, analytical model of the electric field, charge and capacitance in ... more We present a physically based, analytical model of the electric field, charge and capacitance in the collector of a III-V HBT. The model consistently takes into account the decrease of electron drift velocity versus electric field at high fields. We achieve excellent prediction of collector-base capacitance as a function of collector-base voltage and current in the regime of partial depletion

2007 IEEE Bipolar/BiCMOS Circuits and Technology Meeting, 2007

We investigate the relevance of the distribution of the extrinsic collector resistance of a bipol... more We investigate the relevance of the distribution of the extrinsic collector resistance of a bipolar transistor with respect to its small signal characteristics. We show that this distribution is relevant to the admittance parameters in general and to the cut-off frequency and unilateral power gain in particular. We present analytical results for y-parameters in terms of elements of the smallsignal equivalent circuit of the Mextram compact model, extended with distributed extrinsic collector resistance. Hence, our results are relevant to compact modeling, bipolar device technology development and characterization and to design of RF power amplifiers.

Abstract:,This document presents the definition of the CMC world stardard

Extended Abstracts - 2008 8th International Workshop on Junction Technology (IWJT '08), 2008

The possible use of aluminum-mediated SPE silicon islands as ultrashallow Al-doped emitters in PN... more The possible use of aluminum-mediated SPE silicon islands as ultrashallow Al-doped emitters in PNP SiGe HBTs is evaluated by electrical characterization of Si PNP test structures and 1-D MEDICI device simulations. The Al-doped emitters display electrical properties similar to B-doped Si. The surface recombination velocity at the Al-metal to SPE-Si interface and the minority carrier lifetime have been determined to be in the ranges of 7×10 5 -1.2×10 6 cm/s and 2-3×10 -8 s, respectively.

We present two complementary, analytical calculations of the electric field, stored charge, capac... more We present two complementary, analytical calculations of the electric field, stored charge, capacitance and transit time of the collector of a III-V HBT, for respectively low- and high bias regimes. Voltage, current and temperature dependence of base-collector capacitance as well as f/sub T/-peaking at high bias is thus quantitatively related to GaAs-like electron drift-velocity characteristics. The physical insight developed serves as a basis for extension of the Mextram compact bipolar transistor model for circuit simulation.

Journal of Marine Science: Research & Development, Jun 27, 2015

Axioms, Dec 27, 2022

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

We extend the category of parameter extraction methods that utilize weak avalanche under fixed em... more We extend the category of parameter extraction methods that utilize weak avalanche under fixed emitter current conditions to extract base resistance. Our new method consistently accounts for self-heating and Early effect so as to arrive at an accurate extraction of both base and thermal resistance. It involves only small variations in bias and temperature conditions, so that parameter extraction as a function of bias and temperature is enabled. The method is directly applicable to transistors - no dedicated test structures are needed - so that the method is cost effective. The method is demonstrated on RF SiGe HBTs.

We enhance Frobenius' method for solving linear ordinary differential equations about regular sin... more We enhance Frobenius' method for solving linear ordinary differential equations about regular singular points. Key to Frobenius' approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel's equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order.

Legendre’s equation is key in various branches of physics. Its general solution is a linear funct... more Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.

Abstract—As the demands for high operating frequency and high output power of modern bipolar tran... more Abstract—As the demands for high operating frequency and high output power of modern bipolar transistor circuits increase, designers are trying to exploit transistor operating regions where they would be able satisfy both conditions, namely working within an avalanche region. For such applications, compact models have to be at their best, so the designers could have a full overview of the device behaviour and possibilities. In order to accomplish that, there should be an adequate way in characterization of avalanche in the RF regime, as well as in verification of the devices against some figures of merit. This paper gives an insight on possible RF avalanche characterization techniques. Repercussions of working in the avalanche regime on some important transistor properties like unilateral power gain, the maximum avaliable gain and the stability factor is demonstrated. Measurements of modern industrial devices, as well as simulations using the world standard bipolar compact model Mex...

Mextram 504 is the new version of the Philips compact model for bipolar transistors. This documen... more Mextram 504 is the new version of the Philips compact model for bipolar transistors. This document contains the derivation of all the equations that are part of the model. This includes the description of the equivalent circuit of Mextram, the equations for the currents and the charges, the temperature-scaling model, the way self-heating is handled, and the noise model. We then discuss some small-signal approximations and the basis of geometric scaling. The formal model definition needed for implementation is given in another report NL-UR 2000/811 [ 1], parameter extraction is discussed in the report NL-UR 2001/801 [ 2]. These reports and the source code can be found on our web-site [ 3]. c ©Koninklijke Philips Electronics N.V. 2002 iii NL-UR 2002/806— March 2002 Model derivation of Mextram 504 Unclassified report Preface The Mextram bipolar transistor model has been put in the public domain in Januari 1994. At that time level 503.1 of Mextram was used within Koninklijke Philips Ele...

Absfroct- We pment two complementary, analytical calculatlons of the electric field, stored charg... more Absfroct- We pment two complementary, analytical calculatlons of the electric field, stored charge, capacitance and transit time of the collector of a IU-V HBT, for respectively low- and high bias regimes. Voltage, current and temperature dependence of basesollector capacitance as well as Jr-peaking at high hips is thus quantitatively related to GaAs-like electron drift-velocity E characteristics. The physical insight developed will serve as a basis for extension of the Mextram compact bipolar transistor model for circuit simulation. j -- 'g a 10

In this paper an extensive verification of the extraction method (published earlier) that consist... more In this paper an extensive verification of the extraction method (published earlier) that consistently accounts for self-heating and Early effect to accurately extract both base and thermal resistance of bipolar junction transistors is presented. The method verification is demonstrated on advanced RF SiGe HBTs were the extracted results for the thermal resistance are compared with those from another published method that ignores the effect of Early effect on internal base-emitter voltage and the extracted results of the base resistance are compared with those determined from noise measurements. A self-consistency of our method in the extracted base resistance and thermal resistance using compact model simulation results is also carried out in order to study the level of accuracy of the method. Keywords—Avalanche, Base resistance, Bipolar transistor, Compact modeling, Early voltage, Thermal resistance, Self-heating, parameter extraction.

This document presents the definition of the CMC world stardard model Mextram, for vertical bipol... more This document presents the definition of the CMC world stardard model Mextram, for vertical bipolar transistors. The goal of this document is to present the full definition of the model, including the parameter set, the equivalent circuit and all the equations for currents, charges and noise sources. Apart from the definition also an introduction into the physical background is given. We have given also a very basic parameter extraction procedure. Both the background and the parameter extraction are documented separately in dedicated documents. The transition from Mextram 503 to Mextram 504 is described, to enable the translation of a 503 parameter-set to a 504 parameterset. At last we have given some numerical examples that can act as a test of implementation. Delft University of Technology iii June 16, 2009 Mextram version 504.8 Mextram definition document Preface October 2004 The Mextram bipolar transistor model has been put in the public domain in Januari 1994. At that time leve...

AIMS Mathematics

Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the und... more Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the underlying spherical symmetry van der Toorn, Ramses Abstract: We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly, why the phase velocity of these wave modes is insensitive to their orientation with respect to the axis of rotation of the planet, while at the same time the very rotation of the planet is a precondition for the existence of the waves. As we show, answers to both questions can be rooted in Lie group and representation theory. In our study the Rossby-Haurwitz modes emerge in a coordinate-free, as well as in a Ricci tensor rank-free manner. We find them with respect to a continuum of spherical coordinate systems, that are arbitrarily oriented with respect to the planet. Furthermore, we show that, in the same sense in which the Lie derivative of Ricci tensor fields is rankfree, the wave equation for Rossby-Haurwitz modes is rank-free. We find that, for each irreducible representation of so(3), there is a corresponding sufficient condition for existence of Rossby-Haurwitz modes as solutions that are separable with respect to space and time. This condition comes in the form of a system of equations of motion for the coordinate systems. Coordinate systems that move along with Rossby-Haurwitz modes emerge as special cases of these. In these coordinate systems the waves appear as stationary spatial fields, so that the motion of the coordinate system coincides with the wave phase propagation. The general solution of the existence condition is a continuum of moving spherical coordinate systems that precess about the axes of the Rossby-Haurwitz modes. Within a single irreducible representation of so(3), the waves are dispersionless.

We adapt the Mextram 504 compact model for Si and SiGe bipolar transistors to make it suitable fo... more We adapt the Mextram 504 compact model for Si and SiGe bipolar transistors to make it suitable for compact modeling of GaAs HBT's. We discuss the different physics included in the new model, Mextram 3500, and demonstrate the capabilities of the new model on GaAs HBT characteristic simulations. We also show an example of advanced GaAs PA-circuit simulations that have

Gallium Arsenide Integrated Circuit Annual IEEE Symposium, 2003

We present a physically based, analytical model of the electric field, charge and capacitance in ... more We present a physically based, analytical model of the electric field, charge and capacitance in the collector of a III-V HBT. The model consistently takes into account the decrease of electron drift velocity versus electric field at high fields. We achieve excellent prediction of collector-base capacitance as a function of collector-base voltage and current in the regime of partial depletion

2007 IEEE Bipolar/BiCMOS Circuits and Technology Meeting, 2007

We investigate the relevance of the distribution of the extrinsic collector resistance of a bipol... more We investigate the relevance of the distribution of the extrinsic collector resistance of a bipolar transistor with respect to its small signal characteristics. We show that this distribution is relevant to the admittance parameters in general and to the cut-off frequency and unilateral power gain in particular. We present analytical results for y-parameters in terms of elements of the smallsignal equivalent circuit of the Mextram compact model, extended with distributed extrinsic collector resistance. Hence, our results are relevant to compact modeling, bipolar device technology development and characterization and to design of RF power amplifiers.

Abstract:,This document presents the definition of the CMC world stardard

Extended Abstracts - 2008 8th International Workshop on Junction Technology (IWJT '08), 2008

The possible use of aluminum-mediated SPE silicon islands as ultrashallow Al-doped emitters in PN... more The possible use of aluminum-mediated SPE silicon islands as ultrashallow Al-doped emitters in PNP SiGe HBTs is evaluated by electrical characterization of Si PNP test structures and 1-D MEDICI device simulations. The Al-doped emitters display electrical properties similar to B-doped Si. The surface recombination velocity at the Al-metal to SPE-Si interface and the minority carrier lifetime have been determined to be in the ranges of 7×10 5 -1.2×10 6 cm/s and 2-3×10 -8 s, respectively.