High-frequency noise modeling of Si(Ge) bipolar transistors

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H

IGH-FREQUENCY

NOISE MODELING of Si(Ge)

BIPOLAR TRANSISTORS

Francesco Vitale

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High-frequency noise modeling

of Si(Ge) bipolar transistors

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op dinsdag 18 maart 2014 om 15:00 uur

door

Francesco VITALE

Master of Science in Electrical Engineering van Università degli Studi di Napoli “Federico II”, Italië,

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Prof. dr. ir. A. W. Heemink Copromotor:

Dr. ir. R. van der Toorn

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter Technische Universiteit Delft Prof. dr. ir. A. W. Heemink, promotor Technische Universiteit Delft Dr. ir. R. van der Toorn, copromotor Technische Universiteit Delft Prof. dr. ir. G. Jongbloed Technische Universiteit Delft Prof. dr. J. R. Long Technische Universiteit Delft Prof. dr. W. H. A. Schilders Technische Universiteit Eindhoven Prof. dr. N. Rinaldi Università di Napoli “Federico II”, Italië Dr. D. B. M. Klaassen, NXP Semiconductors, Eindhoven

Francesco Vitale,

High-frequency noise modeling of Si(Ge) bipolar transitors, Ph.D. Thesis, Delft University of Technology,

with summary in Dutch.

Keywords: noise, modeling, compact model, bipolar transistor, parameter extraction,

non quasi-static effects, circuit simulation, base resistance.

ISBN: 978-94-6108-615-0

Copyright c 2014 by Francesco Vitale

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“Sono appena arrivato. Piove.”

“Non ti preoccupare. Presto tornerà a splendere il sole.”

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6 Compact noise model implementation 75 6.1 Introduction . . . 75 6.2 Non-quasi-static model . . . 77 6.3 Noise model . . . 79 6.3.1 Model derivation . . . 79 6.3.2 Model implementation . . . 82 6.4 Verilog-A code . . . 84 6.5 Model veriﬁcation . . . 86 6.6 Conclusions . . . 90 7 Discussion 91 7.1 Summary and discussion . . . 91

7.2 Recommendations . . . 95

A Noise representations 97 A.1 The representations . . . 97

A.2 The transformations . . . 98

A.3 Derivation of Fmin, Rn and Yopt . . . 99

B Transmission-line analogue 103 B.1 The analogue description . . . 103

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Contents

B.1.1 Homogeneous case . . . 103 B.1.2 Inhomogeneous case . . . 107 B.2 Some ﬁnal remarks . . . 109

C Noise densities configurations 111

C.1 Indefinite admittance matrix . . . 111 C.2 Indefinite admittance noise PSD matrix . . . 112 C.2.1 Indefinite Van Vliet equation . . . 114

D Mextram parameter set 115

References 117

Summary 125

Samevatting 127

List of publications 129

Acknowledgments 131

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chapter 1

INTRODUCTION

Synopsis – This chapter aims to give an overview of noise modeling issues

related to bipolar transistors. The silicon-germanium heterojunction bipo-lar transistor (SiGe HBTs) technology is briefly introduced in the context of current radio-frequency (RF) analogue applications. Thereafter a review of existing noise analysis techniques for bipolar transistors is presented. Motiva-tion for high-frequency noise modeling and the thesis outline are profiled at the end of the chapter. Flicker noise is not part of this thesis, hence it will not be mentioned in this introductory chapter.

1.1 SiGe HBTs for integrated RF applications

The tremendous growth of the cellular phone market in recent years has drawn more interest in high-frequency analog electronics and guided research efforts towards new costly-efficient and high-performance technological solutions. One of the key building blocks in an RF system is the low-noise amplifier (Fig. 1.1), which receives a variety of signals coming from the antenna and must amplify them without reducing the signal-to-noise ratio significantly. Traditionally, low-noise amplifiers have been implemented exclusively in III-V technologies (i.e. III-V HEMTs, III-V MESFETs), banishing silicon to a central role only in digital electronics. Because of their excellent material properties, III-V semi-conductors have been so far the workhorse of the microwave industry. However the use of compound semiconductor materials suffers from some limiting fac-tors like low integration capabilities, high manufacturing costs and process complexities. Although silicon is not the ideal material for microwave appli-cations, an RF Si-based technology, offering performance comparable to III-V technologies, would allow to circumvent some of the aforementioned difficul-ties encountered in III-V materials, with the additional advantage of a bet-ter thermal management by virtue of the higher thermal conductivity of Si. In the early 1980’s, research efforts focused on the SiGe (silicon germanium) compound, with the aim of using this material for heterostructures, similar to what was done with III-V technologies. From a theoretical point of view, it was known that the introduction of germanium in the base layer of a silicon bipo-lar transistor greatly improves operating frequencies, current, noise and power capabilities, keeping the key advantages of the silicon processing. However the relatively large lattice mismatch between silicon and germanium (greater than 4%) was the main reason why the first practical attempts in making heterojunc-tions in silicon technology failed in the beginning. It was between 1980’s-1990’s that improvements in epitaxial techniques made possible the realization of very

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Fig. 1.1–Simplified schematic representation of a receiver system.

thin, strained, SiGe layers on top of silicon crystals without the generation of misﬁt dislocations [7], which would degrade the transistor properties, making the usage of the SiGe compound useless.

The possibility of growing high quality thin SiGe layers on a Si substrate stimulated further research on SiGe HBTs and many related applications have been demonstrated in normal as well as extreme environment conditions [17, 18, 31, 32]. Nowadays, SiGe HBTs with outstanding high-frequency and low-noise performance are available for volume production, competing with state-of-the-art III-V devices. Their main application includes low-noise amplifiers (LNAs) and low-power microwave amplifiers (PAs), in combination with MOS transistors used for logic and memory [10,16,19,30,69]. Such technology, known as BiCMOS, offers the possibility of combining analog and digital components in a new single chip (and cost effective) architecture: a revolution in analog electronics similar, in a way, to the digital one occurred in 1970’s.

1.2 Noise in semiconductor devices

The operation of semiconductor devices is based on the motion of free carriers in the conduction and valence bands, subject to the eﬀect of an applied external force and of the interaction with other carriers or lattice perturbation. Accord-ing to classical theory [98] the motion of free carriers can be studied in the

context of Brownian motion1_{, with each particle showing velocity ﬂuctuations}

while drifting in a certain direction.

Electrical noise at the device terminals can be viewed as a macroscopic eﬀect induced by ﬂuctuations occurring inside the semiconductor device, also termed as “microscopic noise sources”. Fluctuations of carrier velocity, due to the Brownian motion of free carriers, are related to what is known as

diffu-sion noise. Another noise mechanism is associated with ﬂuctuations in the number of free carriers (recombination noise), due to generation-recombination (GR) processes. These are the two main noise mechanism within semiconductor devices. Since an extremely high number of carriers is involved, such ﬂuctuations are assumed to be known only in a statistical sense. The

1

A complete analysis of Brownian motion can be found in the classical paper by Chan-drasekar [13].

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1.3 Review of existing noise modeling approaches

purpose of physics-based noise modeling approaches is to provide a method to relate macroscopic ﬂuctuations at the device terminal, with the internal microscopic noise sources.

As ﬁnal remark, it has to be reminded that a further noise mechanism is ﬂicker noise: this noise source is particularly important at low frequencies, thus it is not part of this thesis, more devoted to high-frequency noise.

1.3 Review of existing noise modeling approaches

The design and the optimization of electronic systems often requires a detailed knowledge of the inherent noise generated within semiconductor active devices, constituting the core of such systems. In this context modeling of device noise is an essential tool for the optimization of noise performance at both technolog-ical level (device design) and circuit level (circuit design). In practice, however, to account for noise performance optimization in a device design cycle, and es-pecially get a good idea about the relation between device process and device noise performance, is challenging and one must rely on predictive noise simu-lation techniques, so as to minimize dependency on lengthy and costly noise measurements.

Noise characterization forms the subject of an extensive branch of the bipo-lar transistor literature. A common methodology in any noise analysis tech-nique consists of two-steps. First the fundamental noise sources within the semiconductor device have to be determined. Second, the noise parameters of the complete device have to be calculated as the transfer from the internal noise sources to the device terminals. It follows that an accurate prediction of the noise performance of a bipolar transistor depends on both a correct modeling of the intrinsic device and on a good description of all its parasitics as well (extrinsic device).

In the following subsections we group existing noise analysis techniques in three different categories, each one addressing the problem of noise at a differ-ent level of abstraction. Such categories are device noise simulations (only briefly reviewed since not a central topic in this thesis), noise analysis

tech-niques and compact noise model approaches. The ﬁrst two approaches aim at noise analysis and optimization of single devices, while compact models are more circuit-oriented.

1.3.1 Device noise simulations

Device noise simulations are based on the numerical solution of partial differen-tial equations (PDEs) describing the transport of charge carriers in semiconduc-tor devices. In order to simulate the noise characteristics at device terminals, PDE-based models, such as the full hydrodynamic (HD), the energy balance (EB) or the drift-diffusion (DD) models, are provided with a stochastic forcing term known, according to common definition (e.g. see [23]), as the Langevin

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term. In other words, the PDE physical model describing the noiseless device is converted into a Langevin equation by adding the Langevin forcing terms, which account for local microscopic noise sources. The numerical solution of the Langevin-PDEs is carried through the evaluation of the Green’s function, i.e. through the evaluation of the transfer function between any local noise source within the semiconductor region and the speciﬁed external terminal, where the noise characteristics have to be evaluated.

Extensive work has been done by Bonani and Ghione [11, 12] in the di-rection of numerical solution techniques of the stochastic drift-diffusion equa-tions, along the lines suggested in the seminal work of Van Vliet [99]. This approach is based on the assumption of equilibrium conditions, so that the use of fluctuation-dissipations relations, like the Einstein relation, is justified for the evaluation of the local noise sources. In those particular cases where non-stationary transport effects are significant higher-order models are needed: in these models, transport and diffusion noise parameters are calculated by means of full-band Monte Carlo (MC) simulations [43, 44].

These techniques, based on contact-to-contact numerical solutions of the stochastic PDE model, are commonly characterized by a high computational complexity and for this reason can be deployed only as ultimate reference. For noise performance optimization, simpler and more computationally eﬃcient noise analysis techniques still suﬃce in a device design cycle. Such approaches are discussed in the following subsection.

1.3.2 Noise analysis techniques

A common purpose of all the publications about predictive noise modeling techniques is to derive simple analytical formulas which describe the noise parameters of the complete transistor solely in terms of admittance parameters and d.c. quantities like currents, eliminating therefore the need for unwanted noise measurements. It is an established fact that such formula exists at the

intrinsicdevice level [92,99] (cf. Chapter 3). However in the derivation of noise parameters, parasitic elements must be considered as well, with each parasitic resistor contributing with thermal noise. Because of the presence of added resistances, as thoroughly discussed by Polder and Baelde [67], the derivation of analytical noise expressions at the external leads becomes impossible, unless some restrictive assumptions are made.

Most used noise models in the past have been Hawkins’ [34] and Van der Ziel and Bosman’s ones [95]. In both models the noise parameters expressions are derived from a simpliﬁed small-signal equivalent circuits provided with cor-related shot noise sources for base and collector currents, representing the noise model of the intrinsic transistor, and the thermal noise source of the base re-sistance, representing the extrinsic part of the transistor. Later on, Voinigescu et al. [104] and Escotte et al. [21] derived diﬀerent expressions for the noise parameters including also an emitter resistance but neglecting the correlation

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Fig. 1.2–Description of noise sources of a bipolar transistor with the thermo-dynamic method [37] (right-hand side) and with other commonly used methods [21, 34, 95, 104] (left-hand side). Emitter resistance is only included in [21, 104].

between base and collector noise sources. Herzel and Heinemann [37] proposed a different approach in which, contrary to the above-mentioned methods, the derivation of the noise formulas does not rely on any equivalent circuit. This method, known as the “thermodynamic method”, is based on the article [36] of the same authors and it aims to predict the thermal noise of all the resistive elements in a bipolar transistor simply from the real part of the device input admittance (Fig. 1.2). The method has gained some credit in the noise litera-ture, partly due to its appeal and also because in practical cases the final result is as far off the noise measurements as other mentioned methods. However, as already pointed out in [5], its validity seems to be rather questionable. Evi-dence of this is found in the noise expression of a diode: in the low-frequency limit, and in the case of negligible series resistance, the shot noise formula of the diode derived with the thermodynamic method [65] differs by a factor of 2 compared to usual textbook expressions.

Although the use of approximate analytical expressions to predict the noise performance of a complete device directly from measured (or simulated) ad-mittance parameters and d.c. currents is attractive, in practice this is very difficult. The main reason is that, as discussed above, the derivation of such expressions requires oversimplified assumptions compromising the accuracy of final results and even the consistency of the method. In other words, the com-bination of the estimated noise and the measured admittance parameters leads often to unphysical results because the simplified noise model deployed is in-consistent with the admittance parameters of the complete device. Inin-consistent noise models can lead to negative values in the argument of square roots that appear in the calculation of the noise parameters (see Appendix A). This was spotted in particular by Niu et al. [63], which attributed the origin of unphys-ical results to the inconsistency of the intrinsic noise models deployed in the aforementioned methods. However another main reason for inconsistency is

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not only a lack of physics of the aforementioned models2_{due to simplifying}

as-sumptions, but also the fact that the approximate noise expressions, intended to describe the noise behavior of the complete device, are not consistent with the admittance parameters used in the calculations, as touched upon above. Spurred by their argument Niu et al. proposed a new intrinsic noise model (analogous to Rudolph’s model [74]) based on the heuristic and plausible ob-servation that the collector noise current can be modeled as a delayed copy of the emitter noise current. The model, known as “transport noise model”, aims at more accurate noise predictions than conventional noise models by intro-ducing a dedicated noise parameter, the so-called noise delay time, commonly associated with the forward base transit time [42–44, 108]. As such it could be extracted from admittance parameters as a fraction of the base transit time of the diﬀusive charge or directly on noise measurements, if available. The draw-back of the transport noise model resides in the noise delay time, the physical meaning of which seems to be not completely clear and is still subject of critical discussion [35]. Furthermore the transport noise model has no connection with the intrinsic admittance parameters of the bipolar transistor. Hence, this ap-proach may lack guaranteed consistency between the noise and the small-signal model, which, in turn, still can result in inaccuracy in predicted transistor noise performance.

The only viable way to achieve accurate and consistent noise predictions is to rely on a detailed equivalent circuit for the transistor and make sure that all the fundamental noise sources are accounted for in it. The calculation of the noise parameters is then delegated to the circuit simulator, which computes the noise characteristics at a given network port through the evaluation of the transfer function between the local noise source and the specified port [73], in a way conceptually similar to the Langevin method discussed in the previous subsection. This method is consistent by construction since the same transfer function is used by the circuit simulator to calculate the admittance parame-ters. This approach is basically what is used in compact models, described in the next subsection; the difference however is that from a device optimization standpoint it is preferable to have technological instead of electrical parame-ters as inputs in order to ensure a direct link between process technology and RF noise performance. These are indeed the guidelines followed recently by Vanhoucke et al. in [101]. Here a detailed equivalent circuit for the extrinsic transistor was used, while the intrinsic device was modeled by the conventional small-signal circuit, provided with the uncorrelated base and collector noise sources (e.g. as in [104]). Even though this is consistent with the small-signal model of the intrinsic transistor, such model ceases to be valid at higher fre-quencies and moderate current levels due to the omission of intrinsic effects

2

More precisely, noise models discussed by Niu et al., such as Hawkins’ [34], Nielsen [61] and Van der Ziel’s [95] models are actually physics-based but include several simplifying assumptions. Therefore in this case it would be more appropriate to talk about “lack of details” rather than “lack of physics”.

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like correlation and the high-frequency effects related to the distributive na-ture of the minority transport in the base region (this issue will be thoroughly discussed in Chapter 4, 5 and 6). Furthermore the use of a small-signal circuit for the intrinsic transistor is not practical from the point of view of device optimization: small changes in the base doping profile would require a mod-ified parameter set for the small-signal circuit. An improved version of this method [103] replaced the intrinsic small-signal circuit by a one-dimensional TCAD device simulation. This enables more flexibility in terms of intrinsic doping profile optimization. Nevertheless still a simplified uncorrelated noise model for the intrinsic transistor was used in [103] which is, this time, no longer consistent with the small-signal model of the intrinsic transistor.

1.3.3 Compact noise model approaches

In modern circuit design cycles a central role is played by compact models, whose purpose is to describe the current-voltage characteristics of a device in a compact way, i.e. through an equivalent network implementing the model equations with a limited number of model parameters. The most known com-pact model of bipolar transistors is the Spice-Gummel-Poon (SGP) model [26]. Next to it, more advanced compact models have been developed in the past. These 1are: Mextram [1, 91], HiCUM [78] and VBIC [57].

The accuracy of compact model noise predictions , once again, depends on a correct modeling of the device at both the extrinsic and intrinsic level. Noise modeling at the extrinsic level is essentially simple: each parasitic resistor and parasitic p-n junction is provided with a thermal and a shot noise source re-spectively. In order to get a good description of the transistor structure, hence a good transfer of the local noise sources to the device terminals, a lot of care must be laid in the parameter extraction methodology. Noise modeling at the intrinsic device level is very much related to the accuracy of the intrinsic a.c. model. The core of all the aforementioned compact models is based on the charge control concept [9,25]. Because of the quasi-static (QS) assumption un-derlying the charge control principle, conventional noise models of the intrinsic transistor consist of uncorrelated shot noise sources for the base and the col-lector currents. However the QS approximation holds only at low frequencies

(much lower than the cutoﬀ frequency of the intrinsic device, f ≤ fT/10);

un-der high-speed bias conditions (approaching fT,peak) and at frequencies close to

fT, non-quasi-static (NQS) eﬀects, due to the distributive nature of the

minor-ity carrier transport in the base region, become signiﬁcant. In this regime more advanced noise models for the intrinsic transistor are needed. Contextually the small-signal model must be updated to take into account the same NQS eﬀects and therefore be consistent with the noise model. Any attempt to implement a more advanced noise model based on the conventional quasi-static small-signal charge control model is questionable and inconsistent.

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effects have a major impact on noise parameters: the increase of the input conductance with frequency and the correlation between base and collector noise currents which is related to the excess phase shift of the collector current. A proper modeling of both effects is required to avoid significant systematic deviations of noise predictions from noise measurements [76]. The problem of noise modeling at the intrinsic device level consists therefore of two different tasks: modeling of NQS effects and of the intrinsic correlation.

Several methods have been published aiming at an extension of the QS charge control model. In general two techniques can be distinguished: the first one relies on empirical delay parameters to account for NQS behavior of the stored charge in the quasi-neutral base region (“input NQS effects”) and for the additional delay of the collector current (“output NQS effects”). One of the first contribution in this direction can be found in the work of Te Winkel [85], while further improvements were proposed later by Seitchik et al. [79] and Wu and Lindholm [106]. These methods require additional nodes in the conven-tional QS equivalent circuit of the intrinsic transistor. The second technique is based on the partitioning of the base diffusion charge: in other words a frac-tion of the total base diffusion charge is reallocated between the internal base and collector nodes [22]. The advantage of this method is that it does not require additional nodes in the network; the drawback is that, being based on a first-order approximation, input NQS effects are not taken into account. In the theoretical work of Rinaldi et al. [72] the partitioned-charge-based (PCB) model, Seitchik’s, Te Winkel’s and Wu-Lindholm’s models are compared with analytical solutions of the small-signal transport equations expressed in the form of rational polynomials with infinite terms [71]. One of the main con-clusion is that Seitchik’s model appears to be the most accurate, although in practice also the models of Te Winkel and Wu-Lindholm can be considered adequate. This is because the overall dynamic behavior is also influenced by the parasitics surrounding the intrinsic transistor as well, which partly mask the NQS effects. The PCB model is based on a first-order approximation of admittance parameters hence is not as accurate as the NQS models presented in [79, 85, 106].

Several correlated noise models suited for compact model implementations have been proposed in recent years literature [35, 66, 76, 108]. The model of Paasschens et al. [66] is based on the charge-partitioning concept: as such, it does not include the frequency dependence of the base noise current, which stems from the frequency dependence of the input conductance (input NQS eﬀects) of the intrinsic transistor. An extra term which depends on the square of the angular frequency was added to the equation of the base noise current only for consistency requirements, without any physical basis. An analogous model was presented later by Sakalas et al. [76] based, this time, not on charges but on a partition of the intrinsic forward transit time. The recent model of Xia et al. [108] focuses on the impact of collector-base space charge region (CB SCR) on the noise characteristics of bipolar transistor and more precisely on the

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1.4 Aim and outline of the thesis

collector noise current. The relevance of the CB SCR is discussed by Herricht et al. [35] as well: here it is demonstrated that for SiGe HBTs this effect can be safely neglected, while it may be significant for III-V HBTs. Based on these arguments, corroborated by device simulations, a correlated noise model was presented in [35] which neglects CB SCR effects but accounts for second order effects in the base noise current, improving former correlated noise models like [66] and [76].

As we shall discuss in Chapter 6, practical implementation of both corre-lated noise and NQS eﬀects in circuit simulation contexts (e.g. Verilog-A [3]) is not straightforward and, for this reason, it has been the subject of several publications [41, 55, 56, 76]. An accepted common methodology is to use ad-junct networks for the implementation of both correlated noise and NQS eﬀects. Moreover, reliable compact model noise predictions can be achieved only with a proper set of extracted model parameters and, to this scope, the extraction of a proper value for the base resistance is of primary importance.

1.4 Aim and outline of the thesis

The purpose of this thesis is to present improved noise modeling techniques in the context of both device optimization and compact models.

The aim of Chapter 2 is to provide an experimental demonstration of the relevance of the extrinsic network to simulated noise characteristics, as dis-cussed in Subsection 1.3.2. In particular the impact of the distribution of the base resistance is assessed. A method is then presented for the extraction of such distribution from measured admittance parameters.

Next, the study moves into the intrinsic part of the transistor. Chapter 3 gives an insight into basic theory of high-frequency noise in intrinsic bipolar devices. Most seminal works of Van der Ziel, Polder-Baelde and Van Vliet are presented with the aim of giving a theoretical background for subsequent chapters and for a better understanding of the problems of noise modeling.

Chapter 4 presents a novel noise modeling technique suited for device op-timization and characterization. This technique is based on a lumped network derived from a discretization of the PDEs describing the minority-carrier trans-port in QN regions. The proposed approach aims at an improvements of exist-ing techniques (see subsection 1.3.2) which lack in a proper noise modelexist-ing of the intrinsic transistor. The proposed approach is veriﬁed against noise mea-surements of an industrial SiGe HBT. By construction, the lumped network links directly technological parameters with RF noise performance. To explore these capabilities, the impact of intrinsic base doping proﬁle on transistor noise parameters is assessed.

In Chapter 5 a comparative analysis of compact noise model formula-tions for bipolar transistor is presented. The analysis includes the approxi-mated transport noise model, suited for compact model implementations, and

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a physics-based noise model derived systematically from the NQS and the noise theory. In the last part of the chapter a study on the impact of NQS eﬀects originating in the quasi neutral emitter is also presented.

Chapter 6 focuses on the implementation of a uniﬁed correlated noise model, including excess noise due to avalanche multiplication, in the standard compact model Mextram. In order to ensure consistency between noise and small-signal models, a higher-order NQS model is implemented as well. Problems and methods of implementation of both higher-order NQS model and correlated noise in compact model contexts are discussed throughout the chapter.

Chapter 7 includes a conclusive discussion of the main results of this thesis, along with a summary of present state-of-the-art noise modeling technique, and recommendations for possible future developments.

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chapter 2

EXTRACTION OF THE BASE RESISTANCE DISTRIBUTION

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Synopsis – In this chapter we discuss the relevance of the distribution of

the base resistance of planar bipolar transistors with respect to noise and small-signal characteristics. Analytical results for admittance parameters are presented in terms of elements of the small-signal equivalent circuit of the transistor. Next the extraction of the base resistance distribution parameter from measured admittance parameters is discussed for selected cases.

2.1 Introduction

The reliability of noise simulations depends, to a large extent, on the quality of model parameters. Such parameters are in practice extracted from a care-fully chosen set of measured data, preferably kept minimal in terms of required laboratory capabilities and eﬀort. As already emphasized in Chapter 1, with respect to noise modeling and simulation, for example, routine-wise noise mea-surements are preferably avoided in an industrial context: it is highly desirable instead, especially for compact models, to predict the device noise performance on basis of parameters obtained from non-noise measurements. This can be achieved through a detailed equivalent network describing the complete tran-sistor, provided with internal noise sources: the noise at the external terminals is then found in practice by a circuit simulator, through the calculation of the transfer function from the location of the internal noise sources to the device terminals. A fully analogous approach is followed for the evaluation of the y-parameters. Therefore, no additional model parameters are required in order to model the noise characteristics, once the admittance parameters of the complete device have been correctly modeled. As a consequence, by construction the above sketched approach to noise modeling provides a way for prediction of transistor noise performance mainly based on an equivalent circuit-based model, the model parameters of which can be obtained from d.c. and a.c. experimental data only. Hence this approach meets industrial prefer-ences mentioned above.

It is well known that the base resistance is crucial for noise modeling. There-fore the availability of parameter extraction methods for the base resistance is of great practical importance and many publications deal with the subject,

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The content of this chapter was published in: F. Vitale, R. Pijper, and R. van der Toorn, “Base Resistance Distribution in Bipolar Transistors: Relevance to Compact Noise Modeling and Extraction from Admittance Parameters,” In Proc. IEEE Bipolar/BiCMOS

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focusing on extraction of the base resistance from either d.c. (e.g. [102]) or a.c. (e.g. [45]) measurements. However, the literature of base resistance ex-tractions focuses on the extraction of the total base resistance and, to author’s best knowledge, little seems to have been published about the extraction of the

distribution of the base resistance.

For the reason outlined above, in this chapter we address the relevance and extraction of the distribution of the base resistance of bipolar transistors in planar technology. We shall discuss both the relevance of this to noise and admittance parameters data. While the noise aspect illustrates the relevance of correct modeling of the base resistance distribution, understanding of the signiﬁcance of such distribution to admittance parameters may provide a means to extract, or at least verify, distributed base resistance parameters from small-signal measurements.

2.2 Relevance of

R

B

-distribution to noise modeling

Our starting point is the world standard compact model Mextram [1] for Si(Ge) bipolar transistors. The equivalent circuit that forms the backbone of this model is shown in Fig. 6.1. As emphasized in the caption of Fig. 2.1, in

Mextram the total base resistance RB is splitted in a constant part RBc, which

is dependent only on the temperature, and a bias-dependent part represented

by a current source IB1B2. The relevance of the base resistance distribution to

noise modeling is straightforwardly demonstrated in Fig. 2.2. This ﬁgure shows the results of Mextram based simulations of the minimum noise ﬁgure, for two

values of the the RB-distribution parameter XRB, which represents the fraction

of RB assigned to the extrinsic part, corresponding to RBc(XRB = RBc/RB).

The simulated results are plotted against measured data from a QUBiC4G n-p-n SiGe HBT with an emitter area of 0.5µm × 20.3µm [20].

The effect of varying the distribution through XRB, demonstrated in Fig. 2.2 is very similar to the effect of correlation between the intrinsic base and col-lector noise current sources [66, 75]. It can be concluded that the simulations shown in Fig. 2.2 clearly demonstrates the practical relevance of reliable and reasonably accurate extraction of the distribution of the base resistance for noise predictions. Furthermore, Fig. 2.2 suggests that the extrinsic part of the total base resistance gives, effectively, the larger contribution to the minimum noise figure: keeping the total base resistance constant but redistributing it

such that a larger part of it becomes extrinsic, increases Fmin.

2.3 Relevance of

R

B

-distribution to

y-parameters

In this section we focus on the consequences of the distribution of the base resistance for small-signal characteristics (y-parameters). For this purpose, we consider the small-signal circuit shown in Fig. 2.3; this circuit represents

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2.3 Relevance of RB-distribution to y-parameters rE rB rC r C1 rC2 r r r r r C4 r r C3 r r r r r r rE1 r r r r r r r r r r r r r r r B2 B1 r r r r r r S r r r r ❜ ❜ ❜ ❜ CBEO CBCO RE RBc Rc,xx IC1C2 ✒ ♥ ♥IN✻ ✁✁❆❆ IB1 ✁✁❆❆ IB2 QtE QBE QE QS tE ✟✟ ❍❍ IS B1 ♥ ♥Iavl✻ QtC QBC Qepi QB1B2 IB1B2 ✒ ✟ ✠ ✟ ✠ ❆_❆✁✁Iex+IB3 ♥ ♥Isub ❄ ♥ ♥XIsub ❄ Qex Qtex XQtex XQex ❆_❆✁✁ XIex Rc,i Rc,x ♥ ♥ ISf ✻ _Q_t S ✫ ✕ p base n+_emitter n epilayer n+_{buried layer} p substrate

Fig. 2.1–Full equivalent circuit of the Mextram model. The distributed base resistance is represented by two circuit elements: RBc, which is con-stant and depends only on the temperature and the bias-dependent resistance represented by the current source IB1B2.

0 5 10 15 20 0.5 1 1.5 2 2.5 freq [GHz] Fmin [dB] J_C = 0.13 mA/µm2 XRB = 0.7 XRB = 0.3

Fig. 2.2–Mextram based simulation of the minimum noise figure versus fre-quency at VBE = 0.85V and VCE = 2V, for two different values of the RB-distribution parameter XRB. Simulations (solid and dashed line) are compared against measured data (dots) of a QUBiC4G n-p-n SiGe HBT with an emitter area of 0.5µm × 20.3µm [20].

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E B C r C1 rC2 r C4 r C3 rE1 rB2 rB1 S ❜ ❜ ❜ ❜ Re Rbc Rc,xx Repi ♥ ♥ gm·vB2E1 ✻ Cbe hf e0 gm 1 go Cbc,i Rbv Cbc,x Cbc,xx Rc,i Rc,x Cs Rs ✫ ✫ ✕ p base n+ emitter n epilayer n+ buried layer p substrate

Fig. 2.3–Small-signal equivalent circuit of the Mextram model with as used for our small-signal a.c. analysis.

those features of the full equivalent circuit of Fig. 2.1 that are essential to the

problem at hand. As a ﬁrst simpliﬁcation, the charge QB1B2 is neglected. This

is justiﬁed because we shall focus on a.c. measurements for frequencies much

lower than the cutoﬀ frequency of the device considered (fmeas. << fT) and

on SiGe transistors that have a low intrinsic base resistance.

The analytical y-parameters of the circuit shown in Fig. 2.3 can be calcu-lated by means of a step-by-step approach based on the connection of elemental two-ports: this procedure depicted in Fig. 2.4. The admittance parameters are calculated for the common-emitter conﬁguration, with substrate node S

con-nected to the emitter node E. Therefore, if vBE and vCE denote the input

and output small-signal voltages and iB and iC the corresponding input and

output small-signal currents, the admittance parameters are deﬁned as " iB iC # = " y11 y12 y21 y22 # " vBE vCE # . (2.1)

As already mentioned above, the calculation of the y-parameters as function of the elements of the equivalent circuit comes down technically to algebraic manipulation of two-dimensional matrices, the element of which are complex rational functions. In the course of such manipulations, we make use of formal

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2.3 Relevance of RB-distribution to y-parameters

Fig. 2.4–Two-port method for the calculation of the analytical y-parameters of the small-signal circuit shown in Fig. 2.3.

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Taylor expansions in the angular frequency ω, in order to reduce the algebraic complexity of the intermediate and ﬁnal expressions obtained. This means that the result presented here must be interpreted as Taylor expansions about

ω = 0 and, hence, are approximate results on a formal low-frequency limit.

Furthermore, again to reduce the algebraic complexity, we apply the following

limits: hF E0 → ∞ and go → 0, where hF E0 is the common-emitter current

gain and go the output small-signal conductance. Many of the results thus

achieved have been published earlier in [90]. Here we focus on results for the

real part of y12, relevant to the issues at hand.

As a ﬁrst result, we mention that to the leading order in the expansion

in ω, Re(y12) is of order O(ω2). This neatly corresponds to the slope of two

decades per frequency decade as observed in measured data (see Fig. 2.5). In

spite of the simpliﬁcations adopted so far, the full analytical result for Re(y12)

is very complicated. A simpliﬁcation can be achieved in the low-current limit

(gm → 0). At this level of approximation it turns out that Re(y12) is still

aﬀected by collector-substrate coupling eﬀects. These, however, occur through the coupling of the collector resistance and the collector-substrate capacitances.

In the full expression of Re(y12) such contributions tend to be negligible with

respect to terms that represent the coupling of base resistances to emitter-base

capacitance. This observation justiﬁes taking the simplifying limit CS → 0. If,

ﬁnally, the collector plug resistance Rc,xxcan be neglected altogether, we ﬁnd:

− Re(y12)ω−2= Cbc,i(Cbc,i+ Cbe,i)Rb (2.2)

+ Cbc,x(2Cbc,i+ Cbc,x+ Cbe,i)Rbc

+ Cbc,x(2Cbc,i+ Cbc,x)Rc,x+ Cbc,i2 Rc (2.3)

where

Rc= Rc,x+ Rc,i+ Repi (2.4)

Rb= Rbc+ Rbv. (2.5)

In practical applications, it is often not unreasonable to assume that the base-emitter capacitance is dominant over the collector-base capacitance, while the parasitic base resistance is greater than the resistance of the buried layer. Under this assumption, one may neglect the third term of the right-hand-side

of (2.2) with respect to the second; formally, this is done by setting Rc,x→ 0.

From (2.2) and (2.5) we then ﬁnd the distribution parameter:

XRB = −Cbc,i(Cbe,iRb+ Cbc,i(Rb+ Rc)) + Re(y12)ω

−2

Cbc,x(2Cbc,i+ Cbc,x+ Cbe,i)

(2.6)

with XRB = Rbc/Rb. A result slightly more complicated, but very similar to

(2.6), is found without the assumption Rc,x→ 0. Therefore such assumption

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2.4 Experimental verification 109 1010 1011 10−6 10−5 10−4 10−3 10−2 freq [Hz] y 12 [A/V] real imag J C = 0.13 mA/µm 2 V CE = 2 V

Fig. 2.5–Measured (symbols) and Mextram simulated (solid lines) y12versus frequency, at VBE= 0.85V and VCE= 2V.

the course of a parameter extraction procedure, once (I) the base-emitter and base-collector capacitances, (II) the geometrical distribution of the latter and (III) the total base and collector resistances are known, the value of XRB can

be determined in principle from Re(y12). In practice, the total capacitances

are straightforward to extract from measured data, while their geometrical distribution can be found from consideration of either the detailed geometrical layout of the device or geometrical scaling properties of the total capacitances. The total values of base and collector resistances can be obtained from d.c.-based methods. For practical parameter extraction we conclude therefore from

expression (2.6) that the measured value of Re(y12) may serve to extract or

verify the distribution of the base resistance XRB, e.g. by ﬁtting full model

simulated values of Re(y12) to measured ones, in the low current limit.

2.4 Experimental verification

In this section we confront full model (Fig. 2.1) small-signal simulations with measured a.c. characteristics of a QUBiC4G [20] BNA type SiGe HBT with

emitter area AE = 0.5µm × 20.3µm. The model parameters (cf. Appendix D)

have been obtained by standard parameter extraction techniques, supplemented by the interpretation of measured y-parameter characteristics along the lines sketched in Section 2.3 and in Ref. [90].

Fig. 2.6 gives an overview of measured and Mextram simulated admittance parameters. We show simulation results for XRB = 0.7 and XRB = 0.3.

Fig. 2.7 demonstrates the relative sensitivity of Re(y12) to the fraction XRB.

All other admittance parameter components are much less sensitive to XRB

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intermedi-10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 I C [A] y 11 [A/V] imag real 10−5 10−4 10−3 10−2 10−1 10−5 10−4 10−3 I C [A] y 12 [A/V] real imag 10−5 10−4 10−3 10−2 10−1 10−4 10−2 100 I C [A] y 21 [A/V] real imag 10−5 10−4 10−3 10−2 10−1 10−6 10−4 10−2 I C [A] y 22 [A/V] imag real

Fig. 2.6–Measured (symbols) and Mextram simulated (solid and dashed lines)

y-parameters versus collector current, at 5GHz and VCE= 2V. Sim-ulated data correspond to XRB = 0.7 (solid lines) and XRB = 0.3 (dashed lines).

ate current regime. This is in accordance with analytical results published

earlier [90]. It is well known that Re(y22) depends signiﬁcantly on substrate

coupling eﬀects. These have been modeled in detail in the present study: a dedicated network would have to be connected to the Mextram substrate node

to do so. For this reason Re(y22) is not suitable for the extraction or

veriﬁca-tion of XRB. As is demonstrated by Fig. 2.7, Re(y12) is instead very sensitive

to XRB in the low current regime, so that a value for XRB can be extracted

or veriﬁed on measured Re(y12) independently of the modeling of high current

eﬀects, such as bias dependent diﬀusion charges or even self-heating.

2.5 Conclusions

In this chapter we have discussed the relevance of the distribution of the base resistance of planar bipolar transistors with respect to noise and small-signal characteristics. We have experimentally demonstrated that, keeping the to-tal base resistance constant, the distribution of it with respect to the toto-tal base-collector capacitance has an observable impact on noise characteristics. Therefore an incorrect extraction of this distribution may undermine the

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qual-2.5 Conclusions 10−5 10−4 10−3 10−2 10−1 10−5 10−4 10−3 I C [A] Re(y 12 ) [A/V] XRB = 0.7 XRB = 0.3

Fig. 2.7–Measured (symbols) and Mextram simulated (solid lines) Re(y12) versus collector current, at 5GHz and VCE= 2V.

ity of the noise predictions.

We have presented analytical results for the real part of y12, in terms of

elements of the small-signal equivalent circuit of the Mextram compact model. Such analytical results provide a method for the extraction of the base

re-sistance distribution parameter from measured values of Re(y12) in the low

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chapter 3

NOISE THEORY OF BIPOLAR DEVICES

Synopsis – In this chapter most seminal results of the noise theory of bipolar

devices are presented. This includes: the collective approach proposed by Van der Ziel, the equivalent network approach of Polder and Baelde and finally the analytical and systematic derivation of Van Vliet, based on the solution of the Langevin drift-diffusion equations with the Green’s function method. In the second part of the chapter common approximated noise models, most deployed in the noise literature, are discussed, spotting their limitations.

3.1 Introduction

Classical noise analysis methods are commonly characterized by a two-step ap-proach: the first step concerns the calculation of the spontaneous fluctuation within elemental semiconductor regions. In the second step the response to these fluctuations at specified device terminals is calculated. Noise modeling techniques based on this methodology are usually also termed as “collective approaches”. Hence, once the statistical properties of the microscopic noise sources are known, the main issue is to provide a method which relates the in-ternal fluctuations to the ones observed at the exin-ternal terminals of the device. This relation is essentially linear, since the fluctuations are small enough so that they do not perturb the noiseless steady state of the device. As a conse-quence noise analysis can be naturally carried out in the context of small-signal analysis.

First seminal contributions to the theory of noise in bipolar devices can be found in the classical work of Van der Ziel [92], who also proposed a completely different approach based on corpuscular reasoning [89, 94]. Contrary to collec-tive approaches, the corpuscular approach is based on dividing the carriers which cross the junction in different groups, rather than on their flow in the bulk region. Interestingly, albeit the physical bases are different, the collective and the corpuscular approaches give exactly the same result (in low-injection). However the corpuscular approach has never been widely adopted: the reason is that it is very difficult to extend it to the high-injection case and also be-cause the derivations at higher frequencies lack of a strong physical basis. An ingenious method was proposed by Polder and Baelde [67], based on a circuit network representation of the transport equations. They demonstrated that a complete noise characterization of a bipolar device can be obtained by for-mally assigning Nyquist thermal noise to each conductance and by accounting for shot noise sources, due to injection and collection of minority carriers, at the

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endings of their equivalent network. In the appendix of their paper they also make an attempt to solve the drift-diffusion equations by means of the Green’s function method. In this method the involved PDE model is provided with stochastic forcing terms (Langevin terms), counting for inherent fluctuations of the carrier velocity (diffusion noise) and carrier number (GR noise). The solution of the resultant stochastic PDEs can be calculated once the spectra of the internal fluctuations are known. This approach was used in particular by Van Vliet [99], who solved the problem within a more rigorous mathemat-ical framework. Other classmathemat-ical methods not discussed above, yet worthy to be mentioned, are the field impedance method of Shockley et al. [80] and the transfer impedance field approach proposed by Van Vliet et al. [100]. As thor-oughly discussed in [87, 100], these methods can be considered equivalent to Langevin methods (i.e. Green’s function methods).

Selected noise analysis techniques, which represent the physical background of this thesis, are the subject of the following sections. These are: Van der Ziel’s collective approach [92], the equivalent network of Polder and Baelde [67] and the Green’s function method deployed by Van Vliet [99]. It is important to bear in mind that in these methods the analysis is restricted to the intrinsic device (i.e. only quasi-neutral regions), disregarding the parasitic elements. These can be accounted for by adding parasitic resistors and depletion capacitances surrounding the intrinsic device. In the last part of the chapter most known simpliﬁed noise models are discussed, pointing out their major limitations.

3.2 The collective approach

In this section Van der Ziel’s collective approach [92] is discussed. The treat-ment is based on [92], though the derivation is somewhat diﬀerent, following the analogous theory of transmission line response to independent distributed sources [51,52]. The starting point is the transport equations of minority holes injected in a homogeneously doped n-region, which is identiﬁed with the neutral base region of a common-base p-n-p bipolar transistor:

q∂p ′_{(x, t)} ∂x = − 1 ADp ip(x, t) (3.1a) ∂ip(x, t) ∂x = − A τp qp′(x, t) − qA∂p ′_{(x, t)} ∂t . (3.1b)

Here Dp and τp are the hole diﬀusion coeﬃcient and lifetime respectively, q

is the elementary charge, ip the hole current, p′ the excess hole concentration

and A the cross-sectional area, orthogonal to the current density in this case. These equations are analogous to the diﬀerential equations of a distributed

lossy transmission line with negligible self-inductance, given that qp′_{(x, t) is}

recognized as the voltage v(x, t) along the line and ip(x, t) as the current i(x, t)

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3.2 The collective approach

Fig. 3.1–Transmission line analogue of a bipolar transistor with internal noise sources. Ipeand Ipcrepresent the short-circuited emitter and collec-tor noise currents induced by the internal distributed noise sources Vdand Igr. rewritten as: ∂v(x, t) ∂x = −Ri(x, t) (3.2a) ∂i(x, t) ∂x = −Gv(x, t) − C ∂v(x, t) ∂t (3.2b)

where R = 1/ADp, G = A/τp and C = A represent the line resistance,

con-ductance and capacitance per unit length respectively1_{. Note that the current}

at the endings of the line represent the hole emitter and collector currents, ipe

and ipc respectively. More explicitely: i(0) = ipeand i(W ) = ipc.

The microscopic noise sources are taken into account in the line by means of

independent distributed series voltage νd and parallel current ιgr generators,

as depicted in Fig. 3.1: νd represents the diﬀusion noise due to ﬂuctuating

hole charge density and ιgrrepresents the generation-recombination (GR) noise

current in elemental line elements enclosed between x and x + ∆x. Such noise sources appear in the set of equations (3.2) as ﬂuctuating forcing terms, whose only statistical properties are known:

∂v(x, t) ∂x = −Ri(x, t) − νd(x, t) (3.3a) ∂i(x, t) ∂x = −Gv(x, t) − C ∂v(x, t) ∂t − ιgr(x, t) . (3.3b)

As already mentioned in Section 3.1, noise ﬂuctuations are small enough so that noise analysis can be carried out in the context of small-signal analysis. In the frequency domain, the partial diﬀerential equations (3.2) reduce to ordinary

1

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diﬀerential equations, given by: dV(x)

dx = −R I(x) − Vd(x) (3.4a)

dI(x)

dx = −G V(x) − jωC V(x) − Igr(x) (3.4b)

where V, I, Vdand Igrare the Fourier transforms of v, i, νdand ιgrrespectively.

The problem is now to determine the system response for initial conditions given by the distributed series voltage and parallel current noise sources. As done by Van der Ziel in [92] we will consider short-circuited junctions as bound-ary conditions, i.e. V(0) = 0 and V(W ) = 0. This will allow us to calculate the ﬂuctuating short-circuited currents at the endings of the line (x = 0 and

x = W ), due to the distributed current and voltage noise sources. With

ref-erence to Fig. 3.1, the ﬂuctuating short-circuited currents are the emitter and

collector noise currents deﬁned as Ipe at x = 0 and Ipc at x = W . Since the

problem is described by linear diﬀerential equations, superposition of responses

to the independent sources Igr and Vdcan be used.

Following the method of Lindquist [51, 52], to determine Ipe (or Ipc) one

has to: a) calculate the contribution ∆Ipe(x) (∆Ipc(x)) due only to the noise

sources Vd(x)∆x and Igr(x)∆x located at x, b) determine the limit for ∆x → 0,

c) integrate all the contributions from x = 0 to x = W . Therefore from two-port theory, following steps a), b) and c), one gets:

Ipe= Z W 0 Y12(x) Y11(W − x) + Y22(x)[−I gr(x) + Y11(W − x)Vd(x)] dx (3.5) Ipc = − Z W 0 Y21(x) Y11(W − x) + Y22(x)[Igr(x) + Y22(W − x)Vd(x)] dx (3.6)

where Y11, Y12, Y21 and Y22 are the admittance parameters of the line. For a

uniform line, i.e. uniform base doping concentration, x units long,

Yij= " ₁ Z0coth γx − 1 Z0csch γx −Z10csch γx 1 Z0coth γx # . (3.7)

The quantities Z0= [R/(G + jωC)]1/2and γ = [R(G + jωC)]1/2 represent the

characteristic impedance and the propagation constant of the line respectively. Substituting (3.7) into (3.5) and (3.6), we get:

Ipe= Z W 0 Igr(x)sinh γ(W − x) sinh γW − Vd(x) Z0 cosh γ(W − x) sinh γW dx (3.8) Ipc= Z W 0 Igr(x) sinh γx sinh γW + Vd(x) Z0 cosh γx sinh γW dx (3.9)

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3.2 The collective approach

The power spectra Sipe = hIpeIpe∗ i and Sipc= hIpcIpc∗i, and the correlation

spectrum Sipeipc = hIpeIpc∗i, can be calculated once the statistical properties

of Vd and Igr are known. The PSDs of distributed diﬀusion and GR noise

sources, placed in x and x′_{, can be written as [70, 92, 97]:}

Sνd(x, x′) = hVd(x)Vd∗(x′)i = 4q2 ADp [p′ dc(x) + pn0] δ(x − x′) (3.10) Sιgr(x, x′) = hIgr(x)Igr∗ (x′)i = 2q2_A τp [p′ dc(x) + 2pn0] δ(x − x′) (3.11)

where δ(x − x′_{) denotes the Dirac δ-function, p}′

dc(x) is the d.c. excess hole

concentration in the base region and pn0is the equilibrium hole concentration.

Equation (3.10) was stated by Van der Ziel and then only indirectly proved through considerations on the noise of a junction diode at zero bias (see section III in [92]), while (3.11) was proved from shot noise consideration. Making use of (3.10) and (3.11) we ﬁnally get:

Sipe= 2q2_A τp Z W 0 [ p′ dc(x) + 2pn0] sinh γ(W − x) sinh γW 2 dx + 4q 2 ADp|Z0|2 Z W 0 [ p′ dc(x) + pn0] cosh γ(W − x) sinh γW 2 dx (3.12) Sipc= 2q 2_A τp Z W 0 [ p′dc(x) + 2pn0] sinh γx sinh γW 2 dx + 4q 2 ADp|Z0|2 Z W 0 [ p′dc(x) + pn0] cosh γx sinh γW 2 dx (3.13)

while the cross PSD Sipe,ipc∗ = hIpeIpc∗i is given by:

Sipe,ipc∗ = 2q2_A τp Z W 0 [ p′ dc(x) + 2pn0]sinh γ(W − x) sinh γ ∗_x sinh2γW dx − 4q 2 ADp|Z0|2 Z W 0 [ p′ dc(x) + pn0]cosh γ(W − x) cosh γ ∗_x sinh2γW dx . (3.14) Although the analysis has been carried out in the context of p-n-p bipolar transistors, results (3.12)–(3.14) are valid for n-p-n transistors as well. For small W (narrow base regions) recombination is negligible and the ﬁrst term in the integrals (3.12)–(3.14) vanishes. In this case diﬀusion is the predominant noise mechanism and the correlation at low frequencies is almost equal to −1. Thus far, no assumptions have been made in equations (3.12)–(3.14) con-cerning the voltages across the base-emitter (BE) and base-collector (BC) tions. Assuming a forward biased BE junction and a reverse biased BC junc-tion, the stationary hole concentration in the uniformly doped base region is

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then given by: p′ dc(x) = p′dc,E sinh γ0(W − x) sinh γ0W (3.15) where p′

dc,E = pn0[ exp(VBE/VT) − 1 ] is the d.c. excess hole concentration in

the base region at the emitter side (x = 0) and γ0 is the d.c. propagation

constant of the line. Substituting (3.15) into (3.12)–(3.14) and carrying out the integration one arrives at Van der Ziel’s ﬁnal result,

Sie= 4kT Re(y11cb) − 2qIE (3.16a)

Sic = 2qIC (3.16b)

Sieic∗ = 2kT ycb∗₂₁ (3.16c)

where IE and IC are the d.c. emitter and collector currents respectively and

ycb11= qp′ dc,E kT Z0 coth γW , y cb 21= − qp′ dc,E kT Z0 csch γW

are the common-base input and transfer admittances of the transistor.

Simi-larly to the notation used in [28, 72], ycb

11 and y21cb represent the common-base

input and transfer admittances of the transistor. They can be derived from (3.7): a more detailed treatise on this subject is reported in Appendix B.

For bipolar transistors in common-emitter conﬁguration, equations (3.16) must be transformed yielding [95] (cf. Appendix C):

Sib = 4kT Re(y11ce) − 2qIB (3.17a)

Sic = 2qIC (3.17b)

Sibic∗ = 2kT (yce∗₂₁ − gm0) (3.17c)

where IB is the d.c. base current and gm0 is the low-frequency

transconduc-tance.

3.3 Polder-Baelde equivalent network

The problem of noise in bipolar devices was treated by Polder and Baelde [67] with the aid of an equivalent network derived from a discretization of the drift-diffusion equations. Contrary to the analysis of Van der Ziel, discussed in the previous section, their method is of a quite general nature. For instance, there is no assumption regarding the doping profile in the base region, thus allowing for a built-in drift field E due to an arbitrarily varying doping concentration. Moreover the diffusion constant and the lifetime can be function of the position. Finally the model is not restricted to the 1-D case but it is meant for general 3-D structures. The only assumption that is kept is that injection levels are low compared to majority concentrations and that the recombination rate is proportional to the excess minority concentrations.

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3.3 Polder-Baelde equivalent network

Consider the case of holes injected in a volume V of an arbitrarily doped

n-type semiconductor region (which could be the base region of a p-n-p bipolar

transistor). The equations describing the transport of minority holes in the

n-region, accounting also for internal noise ﬂuctuations, are:

jp = −qDp∇p′+ qµp′E+ qh (3.18a) ∂p′ ∂t = − p′ τp − ∇ · jp q + g (3.18b)

where jp is the hole current density and p′the excess hole concentration in the

base region. All quantities (except q) are, in general, function of the spatial variable r = (x, y, z). Because of the independent diﬀusion noise qh and GR

noise g sources, the current density jp and excess hole concentration p′ will

fluctuate around the noiseless steady-state which satisfies the following d.c. drift-diffusion equations: jp,dc= −qDp∇pdc+ qµppdcE (3.19a) 0 = −pdc− pn0 τp − ∇ · jp,dc q (3.19b)

where the subscript “dc” indicates the stationary value.

Once the statistical properties of h and g are known, the set of equations (3.18) permits the evaluation of the stochastic properties of the additional currents appearing at the emitter and collector leads due to internal noise ﬂuctuations. In particular, as done in the previous section, we are interested in the short-circuited induced emitter and collector currents. This is done by Polder and Baelde [67] with the aid of an equivalent network derived from a discretization of (3.18), rather than following an analytical approach. To this

scope, their treatise starts with the introduction of a new variable vl, through

the following relation: vl = (kT /qpdc) p′. This relation can be derived from a

linearization of Boltzmann relation (cf. Appendix B); it relates the excess hole

concentration p′_{with the voltage v}

lat each point of the base region. Therefore,

making use of the new variable and of (3.19a), the model equations (3.18) can be rewritten as: jp= − q2_D ppdc kT ∇vl+ q jp,dc kT vl+ qh (3.20a) q2_p dc kT ∂vl ∂t = − q2_p dc kT τp vl− ∇ · jp+ g . (3.20b)

Equations (3.20) are then discretized by dividing the n-region in cubes of

points (x, y, z) and edges of length ℓ. The variable vl is then identiﬁed as

the voltage at each point of the so-deﬁned lattice. Each term of (3.20a) and (3.20b) can be readily converted in a speciﬁc network representation. For the sake of simplicity we will focus on the x-direction only; same reasonings hold

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Fig. 3.2–Polder and Baelde equivalent network between adjacent points in the x-direction (noise sources are not shown).

for y- and z-directions. Therefore the current ﬂowing in the x-direction is

given by: ipx= ℓ2jpx, where jpx is the x-component of jp. A somewhat more

complicated equivalent network interpretation is given to the second term at

right-hand side of (3.20a). This term is implemented in [67] by a gyrator Gg

between consecutive lattice points and earth, along with conductances

G+(x, y, z) = Gg(x + 1/2, y, z) (3.21)

G−(x + 1, y, z) = −Gg(x + 1/2, y, z) (3.22)

connected at the input and output port of the gyrator respectively. In this way (3.20) can be turned into the following discretized equations in the x-direction:

ipx(x + 1/2) = −[vl(x + 1) − vl(x)] Gd

+ [vl(x + 1) + vl(x)] Gg+ noise (3.23a)

Cdvl(x)

dt = −Grvl(x) + ipx(x − 1/2)

− ipx(x + 1/2) + noise (3.23b)

where Gd, Gr, Gg and C are deﬁned as:

Gd = q2_ℓD p kT pdc(x + 1/2) , Gr= q2_ℓ3 kT τp pdc(x) Gg = qℓ2 2kT jpx,dc(x + 1/2) , C = q2_ℓ3 kT pdc(x)

The resultant equivalent network is shown in Fig. 3.2.

Up to this point nothing has been said about noise sources and their repre-sentation in the equivalent network. Therefore, the next step is to analyze the noise terms in (3.23) and to make them explicit. Random motion of free carriers

(diﬀusion noise) can be represented with a noise current source ιDbetween, say,

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3.3 Polder-Baelde equivalent network

density of this source can be determined considering the random jumps of free carriers between adjacent volume elements: the eﬀect of each one of these events is treated as full shot noise. Thus, for instance, the number of carriers jumping from point (x, y, z) to (x + 1, y, z) is proportional to the number contained in

the volume element (x, y, z): px→x+1∝ αℓ3pdc(x), where α is a proportionality

constant. Conversely, the number of carriers jumping backward is proportional

to the number in the volume element (x + 1, y, z): px+1→x ∝ αℓ3pdc(x + 1).

The difference is the net flow by diffusion, −Dpℓ2∂pdc/∂x, which also defines

α = Dp/ℓ2. The power spectral density of the noise current ιDis given by the

sum of the shot noise terms related with the random jumps:

SιD = 4q 2_ℓD

ppdc= 4kT Gd. (3.24)

Therefore, diﬀusion noise is represented in the network as thermal noise of the

conductance Gd.

The spectral density SιGRof the GR noise current can be found considering

the random process of electron-hole pair creation and recombination. This correspond to a net d.c. GR noise current,

IGR = qℓ3pdc/τp− qℓ3pn0/τp (3.25)

where ﬁrst term is due to electron-hole pair recombination, while the second to electron-hole pair creation. Since the two processes are random and inde-pendent, one would expect a shot noise behavior, i.e.

SιGR= 2qIGR= 2q

2_ℓ3pdc+ pn0

τp

. (3.26)

In the equivalent network this noise source can be represented with a noise

cur-rent source ιGRbetween each node (x, y, z) of the lattice and earth. Considering

the three conductances connected in parallel between (x, y, z) and earth in the

x-direction (Gr, G+ and G−), it is readily veriﬁed that by formally2 assigning

thermal noise to these conductances the spectral density (3.26) is obtained,

4kT (Gr+ G++ G−) ∆x→0 = 4kT Gr+ ℓ ∂Gg ∂x = 4kT q 2_ℓ3 kT pdc τp − q2_ℓ3 2kT pdc− pn0 τp = 2q2ℓ3pdc+ pn0 τp = SιGR (3.27)

where we made use of (3.19b). Therefore, GR noise is represented in the network as thermal noise of all the conductances connected in parallel between any generic lattice point and earth.

2

High-frequency noise modeling of Si(Ge) bipolar transistors

H

IGH-FREQUENCY

NOISE MODELING of Si(Ge)

BIPOLAR TRANSISTORS

Francesco Vitale

High-frequency noise modeling

of Si(Ge) bipolar transistors

PROEFSCHRIFT

Francesco VITALE

CONTENTS

INTRODUCTION

1.1

SiGe HBTs for integrated RF applications

1.2

Noise in semiconductor devices

1.3

Review of existing noise modeling approaches

1.4

Aim and outline of the thesis

EXTRACTION OF THE BASE RESISTANCE DISTRIBUTION

2.1

Introduction

2.2

Relevance of

R

-distribution to noise modeling

2.3

Relevance of

R

-distribution to

y-parameters

2.4

Experimental verification

2.5

Conclusions

NOISE THEORY OF BIPOLAR DEVICES

3.1

Introduction

3.2

The collective approach

3.3

Polder-Baelde equivalent network