Impact of Correlated RF Noise on SiGe HBT Noise Parameters and LNA Design Implications by Xiaojia Jia A thesis submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Master of Science Auburn, Alabama December 14, 2013 Keywords: SiGe HBT, RF noise, LNA design, technology scaling Copyright 2013 by Xiaojia Jia Approved by Guofu Niu, Alumni Professor of Electrical and Computer Engineering Fa Foster Dai, Professor of Electrical and Computer Engineering Stuart Wentworth, Associate Professor of Electrical and Computer Engineering Abstract This work presents analytical models of SiGe HBT and LNA noise parameters accounting for high frequency noise correlation. The models are verified using measurement data and circuit simulation. The impact of noise correlation is shown to be a strong function of base resistance r b which acts as both a noise source and an impedance. Correlation and r b as impedance have oppo- site e?ects on minimum noise figure NF min , which explains why a widely used NF min model that neglects correlation and r b as impedance agreed with measurements. The agreement, however, does not hold for noise matching source resistance R opt , an important parameter for LNAs. With correlation, noise matching condition is better met for impedance matched LNAs. ii Acknowledgments Foremost, I would like to express my sincere gratitude to my advisor Dr. Guofu Niu, for his patience, assistance and guidance throughout the process of writing this thesis. Without his help this work would not haven been possible. I appreciate his immense knowledge and skills that helped me in all the time of research and this thesis. Besides my advisor, I would like to thank the rest of my thesis committee members, Dr. Fa Dai, and Dr. Stuart Wentworth, for their encouragement, insightful comments, and helpful suggestions. I would like to extend my sincere gratitude to former and current labmates in our SiGe HBT research group. I would like to thank Zhen Li for his patient instructions and constant help that are important to my academic growth. Many thanks also go to Ruocan Wang, Jingshan Wang, Wei-Chung Shih, Jingyi Wang, Zhenyu Wang, Pengyu Li, and Rongchen Ma, for the days and nights we were working together, and for all the fun we have had in the last two years. Also, I would like to thank my friends and everyone else who helped me in completion of this work. Last but not the least, I would like to thank my parents, for their selfless support, under- standing and love that protect me and give me courage to face any di?culties throughout my life. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 SiGe HBT fundamental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Noise in Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Terminal Current Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 SPICE Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Correlation Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Analytical Derivation of Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Noise Parameters of LNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Device Noise Parameters and Relations to LNA Noise Parameters . . . . . . . . 16 4 Impact of Correlation on Device Noise Parameters . . . . . . . . . . . . . . . . . . 17 4.1 Analytical Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Correlation?s Impact on R Device opt . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Correlation?s Impact on F Device min . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Correlation?s Impact on X Device opt . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.5 Correlation?s Impact on R Device n . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.6 Two Roles of r b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 iv 4.7 Correlation and r b Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 LNA Design Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Technology Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A Derivation of Intrinsic Noise Sources? Contributions to i out : i c out , i b out , i rb out , and i Rs out . . . 39 A.0.1 i c out : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.0.2 i b out : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A.0.3 i rb out : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A.0.4 i Rs out : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 B Matlab Code for Noise Parameters of Device Calculation . . . . . . . . . . . . . . . 43 C Matlab Code for Noise Parameters of Matched LNA . . . . . . . . . . . . . . . . . 48 v List of Figures 1.1 Simplified schematic of the LNA consisting of a single SiGe HBT. . . . . . . . . . . 2 1.2 Energy band diagrams of a graded-base SiGe HBT and an Si BJT [25][27]. . . . . . 3 2.1 RF noise sources of a transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 1-D bipolar transistor with collector-base space charge region (CB SCR) e?ect [8]. . 7 2.3 PSDofi b andi c usingcorrelationmodelandSPICEmodelversusFrequency,J C =0.414 mA/?m 2 , ? n =0.651E-12 sec, f T =50GHz. . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 Simplified small signal equivalent circuit of LNA. . . . . . . . . . . . . . . . . . . . 10 3.2 Equivalent circuit of simplified LNA without power source. . . . . . . . . . . . . . . 14 3.3 Equivalent small signal circuit of simplified LNA without noise sources. . . . . . . . 15 4.1 Analytical,simulatedandmeasurednoiseparametersofSiGeHBTwithA E =0.8?20?3 ?m 2 versus J C at V CE =3.3 V, f=5 GHz. . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Device noise resistance R n normalized by 50?. . . . . . . . . . . . . . . . . . . . . 21 4.3 Analytical and measured noise parameters of SiGe HBT device versus frequency at V CE =3.3 V, I c =3.47 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 r b ?s impact on NF min of device versus frequency at V CE =3.3 V, I c =3.47 mA. . . . . 23 4.5 r b ?s impact on R opt of device versus frequency at V CE =3.3 V, I c =3.47 mA. . . . . . . 24 vi 4.6 r b ?s impact on X opt of device versus frequency at V CE =3.3 V, I c =3.47 mA. . . . . . 24 4.7 (a)NF Device min versus J C at 5 GHz; (b)R Device opt versus J C at 5 GHz. . . . . . . . . . . . 25 4.8 F min 1 versus frequency at J C =6.62 mA/?m 2 . . . . . . . . . . . . . . . . . . . . . 26 5.1 R LNA opt ,X LNA opt ,NF LNA min ,andNF LNA ofimpedancematchedLNAversusL E atJ C =0.158 mA/?m 2 , f=5 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 NF min , NF, R opt , X opt , and corresponding L E and I C of LNAs using SPICE model and correlation model versus J C at 5GHz. . . . . . . . . . . . . . . . . . . . . . . . 29 6.1 f T , NF Device min , and r b ?g m versus J C of three technology devices. . . . . . . . . . . . 32 6.2 NF Device min and R Device opt ? L E versus r b ?g m of three technology devices. . . . . . . . . 33 6.3 In LNA design, L E , I C , and J C versus lithography node. . . . . . . . . . . . . . . . 33 A.1 Simplified small signal equivalent circuit of LNA . . . . . . . . . . . . . . . . . . . 39 A.2 Simplified small signal equivalent circuit of LNA with intrinsic current noise source i c . 40 A.3 Simplified small signal equivalent circuit of LNA with intrinsic current noise source i b . 40 A.4 Simplified small signal equivalent circuit of LNA with thermal noise source v rb . . . . 41 A.5 Simplified small signal equivalent circuit of LNA with thermal noise source v Rs . . . . 42 vii List of Tables 4.1 S ibib ?, S icic ?, S icib ?, and S ibic ? of Correlation model and SPICE model . . . . . . . . . 17 6.1 Main Features of Three Technologies . . . . . . . . . . . . . . . . . . . . . . . . . 31 viii Chapter 1 Introduction 1.1 Motivation High frequency noise correlation has been shown to be important for accurate modelling of transistor noise parameters, in particular, minimum noise figure NF min is smaller with correlation [1][2][3][4][5][6][7][8][9]. Recently the impact of noise correlation on LNA design was investi- gated using ADS simulation [10]. However, accurate analytical equations that express transistor and LNA noise parameters in terms of small signal equivalent circuit parameters like cut-o? fre- quency f T , transconductance g m , and base resistance r b have yet to be developed using correlated noise model. The most used analytical transistor noise parameter expressions [11][12][13] are derived with varying degree of approximations using the so-called SPICE noise model, which models base and collector current noises as uncorrelated shot noises. These equations have been widely used in both device technology development [14][15][16] and circuit design [12][17]. Transistor noise parameter expressions were derived in [1] and [7] by introducing correlation. However, the base current noise was still assumed to be shot like, while more recent experimental extractions [6][18][19] and impedance field based analysis of modern SiGe HBTs [20] show a strong and clear frequency dependence in base current noise. The 2qI B shot noise component is a result of emitter minority hole velocity fluctuations, and is not correlated to the collector current noise [21][22]. It is the frequency dependent component of base current noise that correlates with collector current noise. This work aims to develop new expressions using a recent transistor correlation noise model [8][23] that accounts for frequency dependent base current noise. As the primary application is 1 low-noise amplifier (LNA) design, derivation is made directly on a LNA as shown in Fig. 1.1. Transistor results are then obtained as a special case by setting the matching inductances to zero. This allows an easier inspection of the relationship between transistor and LNA noise parameters, as well as how it is a?ected by noise correlation. In particular, the new expressions will be used to investigate how correlation a?ects tran- sistor noise parameters and the nice transistor property of being able to approximately achieve simultaneous LNA noise and impedance matching, which was obtained using the SPICE noise model [12][24]. The Z s in Fig. 1.1 is in general equal to R s , 50?, with a reactance X s =0. Tran- sistor size can be optimized for noise matching source resistance R opt =50?, or real part noise matching. Emitter and base inductors L e and L b can then be adjusted for R in =50?, X in =0, or real part and imaginary part impedance matching. The resulting noise matching source reactance X opt ?X s = 0, or imaginary part noise matching. A logical question is if such property still holds with correlation, and if so, how transistor size and LNA noise figure are a?ected. ?? ?? ??? ????? Figure 1.1: Simplified schematic of the LNA consisting of a single SiGe HBT. 1.2 SiGe HBT fundamental SiGe heterojuction bipolar transistor (HBT) technology is extensively used to develop elec- tronics for low noise operation due to its excellent analog and RF performance. The SiGe HBT 2 technology is the first practical bandgap engineering device realized in silicon and can be inte- grated with the modern CMOS technology. As a result of introducing the graded Ge layer into the base of bipolar junction transistor (BJT), SiGe HBT technology has better performance than traditional Si BJT in DC, RF, and noise performance [25][13]. To illustrate the di?erence be- tween the SiGe HBT and the Si BJT, Fig. 1.2 shows the energy-band diagrams for both the the SiGe HBT and the Si BJT biased identically in forward-active mode. The Ge profile linearly increases from zero near emitter-base (EB) junction to some maximum value near collector-base (CB) junction, and then rapidly ramps down to zero. This graded-Ge results in an extra drift field in the neutral base. The induced drift field accelerates minority carrier transportation, thus it minimizes transit time and increases cut-o? frequency [26]. Because of this advantage and low noise performance, SiGe HBTs have been widely used in commercial and military wireless communication applications. Figure 1.2: Energy band diagrams of a graded-base SiGe HBT and an Si BJT [25][27]. 3 1.3 Thesis Structure The work is organized as follows. Chapter 1 gives the motivation of this work as well as an overview of SiGe HBT technology. Chapter 2 presents thermal noise and intrinsic current noises in semiconductor. Derivations of noise parameters are made in Chapter 3. Small signal equivalent circuit analysis is used instead of linear noise two port analysis to better track how each noise source or equivalent circuit parameter, e.g. r b , enters the final noise parameter expressions. While r b as a thermal noise source is well appreciated, its role as an impedance is often neglected, e.g. in [12][13][28]. In Chapter 4, we verify the model equations with measurement and simulation, and examine how correlation a?ects transistor noise parameters, as well as the two roles of r b . The role of r b as impedance is shown to be important. Chapter 5 discusses LNA design implications, followed by technology scaling discussions in Chapter 6. Chapter 7 concludes the work. 4 Chapter 2 Noise in Semiconductor The operation of semiconductor devices is based on free carriers transportation [29]. From the equivalent circuit and compact modeling stand point, the velocity fluctuation caused by major- ity carrier thermal motion can be expressed by the thermal noises of resistance, and the velocity fluctuation caused by minority carrier thermal motion can be equivalently expressed by the in- trinsic terminal current noises [26][30]. Fig. 2.1 shows the thermal noise sources of resistances at base, emitter and collector terminals respectively as well as the terminal current noises of base and collector. Figure 2.1: RF noise sources of a transistor. 5 2.1 Thermal Noise As described by the Nyquist theorem, the power spectral density (PSD) of thermal noise voltage source of a resistance R is usually given by S vr,vr ? = 4KTR, (2.1) and the PSD of thermal noise current source is S ir,ir ? = 4KT R , (2.2) where K is the Boltzmann constant and T is the standardized noise source temperature 290 K. 2.2 Terminal Current Noise 2.2.1 SPICE Noise Model The base and collector terminal noises are defined as shot noise with the PSDs, 8 > > > < > > > : S i c ,i c ? = 2qI C , S i b ,i b ? = 2qI B , S i c ,i b ? = 0, (2.3) where I B and I C are base and collector DC currents. 2.2.2 Correlation Noise Model Considering various noise physics mechanisms, a correlation noise model was developed in [8]. Fig.2.2illustratesthemodelincludingterminalcurrentnoisesduetominoritycarriervelocity fluctuation and due to additional noises from the collector-base space charge region (CB SCR) transport e?ect. i b1 is the base noise current resulting from minority hole velocity fluctuation at emitter, with a PSD of 2qI B . i c1 is the collector noise current resulting from minority electron velocity fluctuation at base, with a PSD of 2qI C . Both of them are independent with frequency 6 Figure 2.2: 1-D bipolar transistor with collector-base space charge region (CB SCR) e?ect [8]. and uncorrelated with each other. i c1 noise current transfer through the CB SCR becomes i c2 and leads to an extra base current noise i b2 =i c1 -i c2 . Through a series of derivation, the final PSDs of correlation model are [8]: 8 > > > < > > > : S i c2 ,i c2 ? = 2qI C , S i b2 ,i b2 ? = 2qI B +2qI C ! 2 ? 2 n , S i c2 ,i b2 ? = j2qI C !? n , (2.4) where !=2?f, ? n is noise transit time which approximately equals to collector transit time [23][8]. Fig. 2.3 shows calculated PSDs of i b and i c versus frequency using correlation and SPICE model. It illustrates that PSDs of i b and i c using SPICE model are constant over frequency, while correlation leads to S ibib ? increase with frequency, and imaginary part of S icib ? decrease with frequency. 7 0 10 20 30 -1 -0.5 0 0.5 1 freq (GHz) Sicic* 0 10 20 30 0.5 1 1.5 2 2.5 x 10 -23 freq (GHz) Sibib* 0 10 20 30 -1 -0.5 0 0.5 1 freq (GHz) real Sicib* 0 10 20 30 -1.5 -1 -0.5 0 x 10 -22 freq (GHz) imag Sicib* Correlation SPICE 2qI B 2qI B +2qI C (wt n ) 2 -j2qI C (wt n ) Figure 2.3: PSD of i b and i c using correlation model and SPICE model versus Frequency, J C =0.414 mA/?m 2 , ? n =0.651E-12 sec, f T =50GHz. 8 Chapter 3 Analytical Derivation of Noise Parameters To obtain insight into device and LNA noise performance for design and optimization, ana- lytical expressions of noise parameters are required. We develope analytical expressions of noise parameters for the LNA in Fig. 1.1. Z s is source impedance that consists of resistance R s and reactance X s . In general, R s =50 ? and X s =0 ?. However, to examine how close noise match- ing is to impedance matching, we need to include X s in Z s to find optimal noise reactance X opt . For general applicability, arbitrary L e and L b are used. Setting L b =0 and L e =0 leads to noise parameters of the transistor. 3.1 Noise Parameters of LNA Fig. 3.1 shows a small signal equivalent circuit of Fig. 1.1. Base-collector capacitance C bc and r ? between base and emitter are neglected for simplicity. This circuit includes two main kinds of RF noise sources of bipolar transistor pointed in the previous chapter, e.g. the terminal resis- tance thermal noise and the intrinsic terminal current noise. v Rs and v rb are thermal noise voltage sources due to R s and r b , respectively. i b and i c are intrinsic terminal current noise sources due to base current and collector current, respectively. C be is capacitance between base and emitter. g m is transconductance. i out is output noise current due to all noise sources: v Rs , v rb , i b , and i c . LNA noise factor, F LNA , is given by: F LNA = 1+ Noise output due to LNA Noise output due to source (3.1) The LNA has noise sources including the thermal noise source v rb due to r b , and the terminal noise currents i b and i c . The power source has a noise source v Rs whose source impedance is Z s . 9 ?? ? ?? ??? ?? ???? ?? ?? ?? ??? ? ? ?? ???????? Figure 3.1: Simplified small signal equivalent circuit of LNA. The noise currents i b and i c are assumed to be correlated with each other, and no correlation is a special condition when i b i ? c and i c i ? b terms are zero. The thermal noise v b is independent to i b and i c . Therefore, (3.1) is rewritten as: F LNA = 1+ D i ic out +i ib out , i ic out +i ib out ? E + ? i rb out ,i rb? out ? ? i Rs out ,i Rs? out ? , (3.2) ? = 1! 2 C be (L b +L e ) +j!C be (Z s +r b ) +j!g m L e , (3.3) i c out = V c be ? g m +i c = j!g m i c L e ? 1 ? +i c , (3.4) i b out = V b be ? g m = g m i b [(Z s +r b ) +j!(L b +L e )] ? 1 ? , (3.5) i rb out = V rb be ? g m = g m v rb ? 1 ? , (3.6) i Rs out = V Rs be ? g m = g m v Rs ? 1 ? , (3.7) where i ic out , i ib out , i rb out , and i Rs out are output noise currents due to i c , i b , r b (v rb ), and R s (v Rs ), respec- tively, and can be obtained using circuit analysis. The detailed analysis is discussed in Appendix A. Noise figure NF relates to noise factor F by NF=10logF. 10 The output noise power produced by i c and i b is: ? i ic out +i ib out , i ic out +i ib out ? ? = Dhic,ic ? i(!C be ) 2 |Z s +r b | 2 +Dhib,ib ? ig 2 m ? (R s +r b ) 2 + (X s + 1 !C be ) 2 Dhic,ib ? ig m !C be ? (Z s +r b ) ? j !C be [j(Z s +r b )] +Dhib,ic ? ig m !C be ? (Z s +r b ) + j !C be [j(Z s +r b )] (3.8) where D = 1 ?? ? = 1 (!C be ) 2 |Z s +r b | 2 +2! T L e (R s +r b ) +! 2 T L 2 e (3.9) where ! T =g m /C be =2?f T , f T is cut-o? frequency. hic,ic ? i=S icic ??f, hib,ib ? i= S ibib ??f, hic,ib ? i=S icib ??f, and hib,ic ? i= S ibic ??f. ?f is a very narrow bandwidth, in which the noise spectral component have a mean square value [13]. S icic ?, S ibib ?, S icib ?, and S ibic ? are PSDs of i b and i c . Substituting hic,ic ? i, hib,ib ? i, hic,ib ? i, and hib,ic ? i into (3.8): ? i ic out +i ib out , i ic out +i ib out ? ? = DS icic ??f(!C be ) 2 |Z s +r b | 2 +DS ibib ??fg 2 m ? (R s +r b ) 2 + (X s + 1 !C be ) 2 D2g m ?f<(S icib ?)(R s +r b ) +D2g m ?f=(S icib ?) ? !C be |Z s +r b | 2 +X s ? , (3.10) where < stands for real part, and = stands for imaginary part. The output noise power produced by r b is: ? i ib out ,i ib? out ? = g 2 m D4kTr b ?f, (3.11) where k is Boltzmann constant, and T is temperature, which is assumed to be equal to standard- ized noise source temperature 290 K here for simplicity. The output noise power produced by R s is: D i R s out ,i R s ? out E = g 2 m D4kTR s ?f. (3.12) 11 Substituting (3.10), (3.11), and (3.12) into (3.2), we get noise figure of an LNA: F LNA =1+ r b R s + S icic ? 4kTR s ? ! ! T ? 2 " ? ! T !g m ` ? 2 + (R s +r b ) 2 # + S ibib ? 4kTR s ? (R s +r b ) 2 +` 2 ? <(S icib ?) 2g m kTR s (R s +r b ) + =(S icib ?) 2kTR s ? ! ! T ?? (R s +r b ) 2 ` ? ! T !g m ` ? , (3.13) where `=X s +!(L b +L e ). In both SPICE model and correlation model, <(S icib ?)=0, thus we set <(S icib ?)=0 in below. To find out the optimum X s , X LNA opt to minimize the noise figure, we solve @F LNA /@X s = 0: @F LNA @X s = S icic ? 2kTR s ? ! ! T ? 2 ? ! T !g m ` ? + S ibib ? 2kTR s ` =(S icib ?) skTR s ? ! ! T ?? ! T !g m 2` ? = 0, (3.14) ` " S ibib ? +S icic ? ? ! ! T ? 2 +2=(S ibib ?) ? ! ! T ? # +S icic ? ? ! ! T g m ? =(S icib ?) g m = 0, (3.15) where ! T =g m /C be . Substituting `=X s +!(L b +L e ) in (3.15), we obtain the expression of X LNA opt : X LNA opt = 1 N ! T g m ! h S icic ? +=(S icib ?) ? ! T ! ?i ! (L b +L e ), (3.16) where N = S icic ? +S ibib ? ? ! T ! ? 2 +2=(S icib ?) ? ! T ! ? . (3.17) To find the optimum R s that minimizes F LNA , R LNA opt , we solve @F LNA /@R s = 0. The result is: R LNA opt = p A N , (3.18) 12 A = r 2 b N 2 + ? ! T ! ? 2 4kTr b N | {z } v rb contribution + 1 g 2 m ? ! T ! ? 4 S icic ?S ibib ? =(S icib ?) 2 , (3.19) Substituting (3.16) and (3.18) into (3.13) leads to the minimum noise factor of LNA, F LNA min : F LNA min =1+ r b R LNA opt |{z} v rb contribution + R LNA opt +r b 2 4kTR LNA opt ? ! ! T ? 2 N + 1 4kTR LNA opt 1 g 2 m ? ! T ! ? 2 S icic ?S ibib ? =(S icib ?) 2 N , (3.20) where the r b /R LNA opt term is due to v rb which represents the e?ect of r b as a noise source. The remaining terms of (3.20) are due to i c , i b , and v Rs , and depend on r b because r b a?ects how i c , i b , and v Rs are transferred to i out . Here r b ?s e?ect is manifested as an impedance element. While r b as a noise source is generally understood to be important, r b as an impedance element is much more important in a?ecting F min , as detailed below in Chapter 4. Noise resistance R LNA n could be obtained the following equation which is from linear noisy two-port theory [13][31]: R n = S v a ,v ? a 4KT , (3.21) where S v a ,v ? a is chain representation equivalent input noise voltage [26][31], and could be calcu- lated by: S v a ,v ? a = ? i out ,i ? out ? Rn |Y 21 | 2 , (3.22) where Y 21 is the forward transfer admittance with output short circuit, and ? i out ,i ? out ? Rn is the total output noise power without power source, as shown in Fig. 3.2: ? i out ,i ? out ? Rn = ? (i ic out,Rn +i ib out,Rn ),(i ic out,Rn +i ib out,Rn ) ? ? Rn + ? i r b out,Rn ,i r b ? out,Rn ? . (3.23) We find out i ic out,Rn , i ib out,Rn ,i r b out,Rn using equivalent circuit in Fig. 3.2: ? Rn = 1! 2 C be (L b +L e ) +j!C be r b +j!g m L e , (3.24) 13 ?? ? ?? ? ???? ?? ? ?? ???? Figure 3.2: Equivalent circuit of simplified LNA without power source. i c out,Rn = i c ? Rn ? 1! 2 C be (L e +L b ) +j!C be r b ? , (3.25) i b out,Rn = g m i b ? Rn [r b +j!(L b +L e )], (3.26) i rb out,Rn = g m v rb ? Rn . (3.27) The output noise power produced by i c and i b is: ? (i ic out +i ib out ),(i ic out +i ib? out ) ? Rn = D Rn hic,ic ? i h 1! 2 C be (L e +L b ) 2 +! 2 C 2 be r 2 b i +D Rn hib,ib ? ig 2 m ? r 2 b +! 2 (L e +L b ) 2 ? 2D Rn ?f<(S icib ?)g m r b +2D Rn ?f=(S icib ?)g m ? !C be r 2 b +! (L e +L b ) 1! 2 C be (L e +L b ) ? , (3.28) where D Rn is: D Rn = 1 ? 1! 2 C be (L e +L b ) ? 2 +! 2 (C be r b +g m L e ) 2 . (3.29) The output noise power produced by r b is: ? i r b out ,i r b ? out ? = g 2 m D Rn ?4kTr b ?f. (3.30) 14 ? ?? ? ? ?? ?? ???? ? Figure 3.3: Equivalent small signal circuit of simplified LNA without noise sources. Y 21 is calculated using Fig. 3.3: Y 21 = i out V s = i c r b i c +V be +j!L b i b +j!(i b +i c )L e , (3.31) 1 Y 21 = r b RF + 1 g m + j! RF (L b +L e ) +j!L e , (3.32) where RF = g m j!C be = ! T j! . Therefore, 1 |Y 21 | 2 = ? 1 g m ! 2 ! T (L b +L e ) 2 + ? r b ! ! T +!L e ? 2 = 1 g m h ? 1! 2 C be (L e +L b ) ? 2 +! 2 (C be r b +g m L e ) 2 i = 1 g 2 m D Rn . (3.33) From (3.21)-(3.33), we obtain the noise resistance of LNA, R LNA n : R LNA n =r b + r 2 b +! 2 (L e +L b ) 2 4KT S ibib ? + ? 1! 2 C be (L e +L b ) ? 2 +! 2 C 2 be r 2 b 4KTg 2 m S icic ? + r b 2KTg m <(S icib ?) + !C be r 2 b ! (L e +L b ) 1! 2 C be (L e +L b ) 2KTg m =(S icib ?). (3.34) 15 Setting <(S icib ?)=0 in (3.34), the noise resistance of LNA is: R LNA n =r b + r 2 b +! 2 (L e +L b ) 2 4KT S ibib ? + ? 1! 2 C be (L e +L b ) ? 2 +! 2 C 2 be r 2 b 4KTg 2 m S icic ? + !C be r 2 b ! (L e +L b ) 1! 2 C be (L e +L b ) 2KTg m =(S icib ?). (3.35) 3.2 Device Noise Parameters and Relations to LNA Noise Parameters An inspection of (3.16), (3.18), and (3.20) shows that L b and L e only enter (3.16), expres- sion of X LNA opt , and 3.35, expression of R LNA n . Therefore, transistor R opt and F min are as the same as LNA R opt and F min : R Device opt = R LNA opt , (3.36) F Device min = F LNA min . (3.37) Setting L b and L e to zero in (3.16), we obtain transistor X Device opt : X Device opt = X LNA opt +! (L b +L e ) = 1 N ! T !g m h S icic ? +=(S icib ?) ? ! T ! ?i . (3.38) Setting L b and L e zero in (3.35), we obtain noise resistance of device R Device n : R Device n = r b + r 2 b 4KT S ibib ? + 2 6 4 1 4KTg 2 m + ? ! ! T ? 2 r 2 b 4KT 3 7 5 S icic ? + ? ! ! T ? r 2 b 2KT =(S icib ?). (3.39) These relations between device and LNA noise parameters can also be obtained using two- port combining techniques [13][31], and hold with or without correlation. 16 Chapter 4 Impact of Correlation on Device Noise Parameters In the SPICE model, i b and i c noise currents are described as shot noise of majority carriers passing through the EB and CB junction, which relate to the corresponding DC currents by 2qI. At higher frequency,i b increases with frequency and is correlated withi c , due to both base and CB SCR transport, with the later dominant in modern HBTs [23]. For analytical analysis, we use the correlation model in [8]. Table 4.1 shows S ibib ?, S icic ?, S icib ?, and S ibic ? of correlation model in [8] and SPICE noise model. ? n is noise transit time which approximately equals to collector transit time [23][8]. Substituting expressions ofS ibib ?, S icic ?, and=(S icib ?) into (3.36), (3.37), (3.38), and (3.39), we obtain device noise parameters with and without correlation, i.e. the SPICE model. Table 4.1: S ibib ?, S icic ?, S icib ?, and S ibic ? of Correlation model and SPICE model SPICE Correlation S icic ? 2qI C 2qI C S ibib ? 2qI B 2qI B +2qI C (!? n ) 2 S icib ? 0 j2qI C (!? n ) S ibic ? 0 j2qI C (!? n ) 4.1 Analytical Model Verification Fig. 4.1 compares analytical model, Agilent ADS simulation and measurement of F Device min , R Device opt , and X Device opt versus J C at 5 GHz. The device used is from a commercial SiGe HBT BiCMOS technology, with an emitter area A E of 0.8?20?3 ?m 2 , a peak f T of 36 GHz, and a peak f max of 65 GHz. Measurement data were obtained using a commercial system, and have been de-embedded. A modified HICUM model with correlation is used [8]. ? n is extracted by fitting measured noise parameters [19]. Other compact model parameters are extracted by fitting 17 DC I-V curves and Y-parameters. For calculation, i b , i c , r b , g m , ! T , and ? n are generated from operation point information using the ddx operator in the Verilog-A device model. The Matlab codes for noise parameters calculation are shown in Appendix B. 0 0.5 1 1.5 0 5 10 15 Jc(mA/um 2 ) NF Device min (dB) 0 0.5 1 1.5 0 20 40 60 Jc(mA/um 2 ) R Device opt (ohm) 0 0.5 1 1.5 0 50 100 Jc(mA/um 2 ) X Device opt (ohm) Analytical,correlation Analytical,SPICE Simulation,correlation Simulation,SPICE Measurement (b) (a) (c) Figure 4.1: Analytical, simulated and measured noise parameters of SiGe HBT with A E =0.8?20?3 ?m 2 versus J C at V CE =3.3 V, f=5 GHz. Analytical model agrees fairly well with simulation. Correlation model is much closer to measured data than SPICE [10], especially at higher J C . Correlation leads to smaller NF min and larger R opt as detailed below. 18 4.2 Correlation?s Impact on R Device opt Since emitter length is first decided in LNA design for R Device opt =50 ?, and NF min requires R opt , we discuss the impact of correlation on R Device opt first. In Fig. 4.1 (b), the R Device opt with cor- relation is larger. To see if this can be generalized, we further simplify R Device opt by substituting expressions of PSDs from Table 4.1 into (3.36), and approximating g m with (qI C )/(kT). The N in (3.17) can be rewritten as 2qI c M, with M being: M = 8 > < > : M Spice = 1+ 1 ! T ! 2 , SPICE M Cor = 1+ 1 ! T ! 2 + (! T ? n ) 2 ! T ? n , Correlation (4.1) where =I C /I B , M Spice and M Cor are M for SPICE model and correlation model. Then, R Device opt = v u u u u u t r 2 b + ? ! T ! ? 2 2 6 6 4 2r b g m M |{z} v rb contribution + 1 M 2 1 g 2 m ? ! T ! ? 2 3 7 7 5 . (4.2) (4.2) showsthattheonlydi?erencebetweenR Device opt withandwithoutcorrelationisfactorM. As f T <1/(2?? f ), and ? n R Device opt,Spice , the second term r b /R Device opt is smaller with correlation. Since M cor R Device opt,cor < > : L e = R s r b ! T , L b = 1 ! 2 C be R s r b ! T . (5.1) Then we will examine how correlation a?ects X LNA opt , which is supposed to approximately be zero using SPICE model [12][24]. Fig. 5.1 shows analytical R LNA opt , X LNA opt , NF LNA min , and NF LNA of impedance matched LNA versus L E at J C =0.158mA/?m 2 , f=5GHz. R LNA opt , X LNA opt , and NF LNA min are plotted by substituting (5.1) into (3.16), (3.18), and (3.20) that impedance matches at each L E . NF LNA are plotted by setting R s =50? and X s =0 in (3.13). The markers on R LNA opt curves are firstly decided at L E required for R LNA opt =50?. Then X LNA opt , NF LNA min , and I C at corresponding L E are also marked. At L E required for R LNA opt =50 ?, X LNA opt equals to -2.924 ? for correlation model and - 8.286 ? for SPICE model. The closeness between X LNA opt and zero is better with correlation. Even though X LNA opt slightly deviates from zero, NF LNA is very close to NF LNA min . Also, noise 27 20 40 60 80 100 120 0 50 100 150 200 X: 42.33 Y: 50.08 Emitter Length L E (um) R LNA opt (ohm) X: 64.46 Y: 50.12 20 40 60 80 100 120 -20 -15 -10 -5 0 5 X: 64.27 Y: -2.924 Emitter Length L E (um) X LNA opt (ohm) X: 42.09 Y: -8.286 20 40 60 80 100 120 1 1.5 2 2.5 3 X: 42.56 Y: 2.015 Emitter Length L E (um) NF LNA & NF LNA min (dB) X: 64.74 Y: 1.39 20 40 60 80 100 120 0 5 10 15 20 X: 64.95 Y: 8.202 Emitter Length L E (um) I c (mA) X: 42.32 Y: 5.344 Correlation SPICE Correlation SPICE NF min,cor NF cor NF min,SPICE NF SPICE X opt =0 SPICE Correlation (a) (b) (c) (d) Figure 5.1: R LNA opt , X LNA opt , NF LNA min , and NF LNA of impedance matched LNA versus L E at J C =0.158 mA/?m 2 , f=5 GHz correlation provides smaller distance between NF LNA and NF LNA min . In this case, correlation leads to more closeness between noise and impedance matching. A larger L E required using correlation model results in a higherI C : 8.202mA for correlation model versus 5.344mA for SPICE model. To constrain power consumption, L E does not have to be chosen exactly at R LNA opt =50?, since slight deviation of X LNA opt barely increases NF LNA min , so does R LNA opt . One can choose a smaller L E like 50?m using correlation model for lower I C . X LNA opt,Spice = ! T ! g m 1 ! T ! 2 h 1+ 1 ! T ! 2 i, (5.2) 28 0.1 0.2 0.3 1 2 3 Jc(mA/um 2 ) NF LNA min (dB) 0.1 0.2 0.3 1 2 3 Jc(mA/um 2 ) NF LNA (dB) 0.1 0.2 0.3 46 48 50 52 Jc(mA/um 2 ) R LNA opt (ohm) 0.1 0.2 0.3 -10 -5 0 5 10 Jc(mA/um 2 ) X LNA opt (ohm) Analytical,Correlation Simulation,Correlation Analytical,SPICE Simulation,SPICE 0.1 0.2 0.3 20 40 60 80 Jc(mA/um 2 ) L E for R opt =50ohm (um) 0.1 0.2 0.3 0 5 10 15 Jc(mA/um 2 ) Ic for R opt =50ohm (mA) (a) (b) (d)(c) (e) (f) Figure 5.2: NF min , NF, R opt , X opt , and corresponding L E and I C of LNAs using SPICE model and correlation model versus J C at 5GHz. X LNA opt,cor = ! T ! g m 1 ! T ! 2 + (! T ? n ) 2 (! T ? n ) h 1+ 1 ! T ! 2 + (! T ? n ) 2 2(! T ? n ) i. (5.3) Then we repeat LNA design at di?erent J C (V be ). Fig. 5.2 shows NF LNA min , NF LNA , R LNA opt , and X LNA opt , and corresponding L E and I C with and without correlation versus J C at 5GHz. The simulation data are generated using a cascode LNA [10][34]. The analytical curves agree well with simulation. In Fig. 5.2, X LNA opt with correlation is relatively closer to zero. X LNA opt without correlation is negative, i.e. (5.2), while X LNA opt with correlation is positive at small J C , which agrees with analytical expressions, i.e. (5.3). 29 At every J C bias, L E is required to be rescaled. Although L E could be chosen by optimizer in some simulator like Agilent ADS, it takes a plenty of time to generate one curve. By contrast, the L E can be easily scaled for R LNA opt =50? in calculation. Thus, analytical approach improves the e?ciency of LNA design. 30 Chapter 6 Technology Scaling To investigate e?ect of technology scaling, we compare three lithography nodes 0.5?m, 0.24?m, and 0.13?m. Parameters of the reference device are given in Table 6.1. Table 6.1: Main Features of Three Technologies Lithography Node (?m) 0.50 0.24 0.13 Peak f T (GHz) 36 60 212 Peak f max (GHz) 100 120 265 Working f (GHz) 6 10 35 Normalized r b *L E (??m) 412.9 341.4 210.8 Emitter area A E (?m) 0.8?20?3 0.24?20?1 0.12?12?1 With increasing frequency, the contribution of 1 in (4.2) and (4.3) is small [11]. (4.2) and (4.3) are simplified by neglecting contribution of 1 except that in M: R Device opt = r b s 1+ 2 r b g m M ? ! T ! ? 2 , (6.1) F Device min = 1+r b g m M ? ! ! T ? 2 2 4 1+ s 1+ 2 r b g m M ? ! T ! ? 2 3 5 , (6.2) where term r b g m is independent of L E for a given J C , because r b / 1/L E and g m /L E . Fig. 6.1 shows f T , NF Device min , and r b ? g m versus J C of three technologies. Because of dif- ferent f T , for a fair comparision, the J C at which f T =f T,peak /2 in each technology is used. Since working frequency f is chosen as f T,peak /6, ! T /!=3 in (6.1) and (6.2). Fig. 6.2 shows NF Device min and R Device opt ? L E versus r b g m . The trends of NF Device min are consistent with r b g m : smaller r b g m leads to smaller NF Device min . In (6.1), R Device opt ? L E is related with not only 31 10 -3 10 -2 10 -1 10 0 10 1 10 2 0 100 200 X: 1.463 Y: 106.1 Jc(mA/um 2 ) f T (GHz) X: 0.1643 Y: 30.95 X: 0.05966 Y: 18.07 0.5 um 0.24 um 0.13 um 10 -3 10 -2 10 -1 10 0 10 1 10 2 1 2 3 4 5 6 Jc(mA/um 2 ) NF Device min (dB) 10 -3 10 -2 10 -1 10 0 10 1 10 2 0 2 4 6 Jc(mA/um 2 ) r b ? g m dash: SPICE solid: Correlation Figure 6.1: f T , NF Device min , and r b ?g m versus J C of three technology devices. r b g m , but also r b ? L E . The 0.13?m technology has much smaller R Device opt ? L E than the others, which will lead to smaller L E in LNA design. Fig. 6.3 shows L E for noise matching and corresponding I C and J C versus lithography node in LNA design. Although J C is highest at 0.13?m node, its smaller L E requested for R LNA opt =50? and smaller W E lead to less I C since I C =L E W E J C . Despite the increasing J C required to enable higher frequency design, the smaller L E and W E required for noise matching help keeping power consumption of LNA low. 32 0.2 0.4 0.6 0.8 1 1.5 2 2.5 r b ? g m NF Device min (dB) 0.2 0.4 0.6 0.8 500 1000 1500 2000 2500 3000 r b ? g m R Device opt L E (ohm*um) SPICE Correlation 0.5 um 0.13 um 0.5 um 0.13 um Correlation SPICE 0.24 um 0.24 um Figure 6.2: NF Device min and R Device opt ? L E versus r b ?g m of three technology devices. 0 0.13 0.24 0.5 25 30 35 40 45 50 55 60 Lithography Node (um) Emitter Length L E (um) 0 0.13 0.24 0.5 0 1 2 3 4 5 LNA I C (mA) Lithography Node (um) 0 0.1 0.2 0.3 0.4 0.5 Jc(mA/um 2 ) Ic,SPICE Ic,Correlation Jc Correlation SPICE Figure 6.3: In LNA design, L E , I C , and J C versus lithography node. 33 Chapter 7 Conclusion The general analytical expressions of SiGe HBT and LNA noise parameters have been de- veloped using small signal circuit analysis and verified by simulation and measurement data. 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IEEE MTT-S International Microwave Symposium Digest, vol. 1, pp. 517?520, Jun 2002. 37 Appendices 38 Appendix A Derivation of Intrinsic Noise Sources? Contributions to i out : i c out , i b out , i rb out , and i Rs out ?? ? ?? ??? ?? ???? ?? ?? ?? ??? ? ? ?? Figure A.1: Simplified small signal equivalent circuit of LNA Fig. A.1 is small signal equivalent circuit of LNA with all noise sources. The transistor noise sources include the terminal current noises i c and i b , and the thermal noises v b of r b . Power source has a noise source v s of Z s . i c out , i b out , i rb out , and i Rs out are denoted as contributions of i c , i b , v b , and v s to total output noise current i out , respectively. Each could be calculated using circuit analysis by removing the other noise sources in Fig. A.1. A.0.1 i c out : i c out could be calculated by circuit analysis using Fig. A.2, which includes only one noise source, i.e. i c . V c be ? j!C be [Z s +r b +j!(L b +L e )]+V c be +j!(i c +g m V c be )L e = 0. (A.1) Then, 39 ? ?? ? ?????? ?? ?? ?? ????? Figure A.2: Simplified small signal equivalent circuit of LNA with intrinsic current noise source i c . V c be = j!i c L e ? 1 ? , (A.2) where ? = 1! 2 C be (L b +L e ) +j!C be (Z s +r b ) +j!g m L e . (A.3) Therefore, i c out = V c be ? g m +i c = j!g m i c L e ? 1 ? +i c , (A.4) A.0.2 i b out : ? ?? ? ?????? ?? ? ??? ? ? Figure A.3: Simplified small signal equivalent circuit of LNA with intrinsic current noise source i b . 40 i b out could be calculated by circuit analysis using Fig. A.3, which includes only one noise source, i.e. i b : V b be = i b [(Z s +r b ) +j!(L b +L e )] ? 1 ? , (A.5) i b out = V b be ? g m = g m i b [(Z s +r b ) +j!(L b +L e )] ? 1 ? . (A.6) A.0.3 i rb out : ?? ? ?? ? ?????? ?? ??? ? ?? Figure A.4: Simplified small signal equivalent circuit of LNA with thermal noise source v rb . i rb out could be calculated by circuit analysis using Fig. A.4, which includes only one noise source, i.e. v rb : V rb be = v rb ? 1 ? , (A.7) i rb out = V rb be ? g m = g m v rb ? 1 ? . (A.8) A.0.4 i Rs out : i Rs out could be calculated by circuit analysis using Fig. A.2, which includes only one noise source, i.e. v Rs : V Rs be = v Rs ? 1 ? , (A.9) i Rs out = V Rs be ? g m = g m v Rs ? 1 ? . (A.10) 41 ?? ?? ??? ?? ? ?? ???? ?? ???? Figure A.5: Simplified small signal equivalent circuit of LNA with thermal noise source v Rs . 42 43 Appendix B Matlab Code for Noise Parameters of Device Calculation close all; clear all; format long; datapath = 'C:\Users\Xiaojia\Documents\Iccap\Noise\Noise5PAE_Hicum\7WLNoise_wrk\export_d ata\'; OPinfo = sprintf('%s7WL_p24_OPinfo.csv',datapath); OPinfo_wt = sprintf('%s7WL_p24_OPinfo_wt.csv',datapath); [OP] = textread(OPinfo,'','delimiter',',','headerlines',46); %'Vbe','0Hz','1','OP_betadc','OP_ib','OP_ic','OP_rb','OP_Cbe','OP_taun' [OP_wt] = textread(OPinfo_wt,'','delimiter',',','headerlines',36); %'','','','wt_gm','','','wt_fT' [Ic] = OP(:,6)'; [rb] = OP(:,7)'; [Cbe] = OP(:,8)'; [beta] = OP(:,4)'; [taun] = OP(:,9)'; [wt] = OP_wt(:,7)'; freq = 10e9; Length_e0 = 1 * 20; Width_e = 0.24; Ae = Width_e * Length_e0; q = 1.602189e-19; twoq = 2 * q; w = 2 * pi * freq; Temp = 290; 44 k = 1.380662e-23; kt = k * Temp; m = size(Ic,2); for n = 1:m Cbe_bc(n) = Cbe(n); Ib(n) = Ic(n)/beta(n); gm(n) = wt(n)*Cbe_bc(n); wCbe_inv(n) = 1/(w * Cbe_bc(n)); wt_w(n) = wt(n)/w; Jc(n) = Ic(n)/Ae; fg1 = 0.757; T(n) = taun(n)*fg1; %%-----------------------------correlation---------------------------- Sib(n) = twoq * Ib(n) + twoq * Ic(n) * (w * T(n)).^2; Sic(n) = twoq * Ic(n); Bu(n) = -twoq * Ic(n) * w * T(n); AA(n) = Sic(n) + Sib(n) * wt_w(n).^2 + 2 * Bu(n) * wt_w(n); Xopt(n) = wCbe_inv(n) * (Sic(n) + Bu(n) * wt_w(n))/AA(n);% - wCbe_inv(n); % "- wCbe_inv(n)" for LNA Ropt2(n) = 4*kt * rb(n) * wt_w(n).^2/AA(n) + rb(n).^2 + wCbe_inv(n).^2 * Sic(n)/AA(n) - wCbe_inv(n).^2 * (Sic(n) + Bu(n) * wt_w(n)).^2/(AA(n).^2); Ropt(n) = sqrt(Ropt2(n)); Fmin(n) = 1 + rb(n)/Ropt(n) + 1/(4 * kt * Ropt(n)) * (1/wt_w(n)).^2 * (Ropt(n) + rb(n)).^2 * AA(n) + 1/(4 * kt * Ropt(n)) * wCbe_inv(n).^2 * (Sic(n) * Sib(n) - Bu(n).^2) / AA(n) ; NFmin(n) = 10*log10(Fmin(n)); R(n) = (50+rb(n)).^2+0.^2; F(n) = 1+rb(n)/50+Sic(n)*(1+(w*Cbe_bc(n)).^2*R(n)- 2*w*Cbe_bc(n)*0)/(gm(n).^2*4*kt*50)+Sib(n)*R(n)/(4*kt*50)- 2*Bu(n)*0/(gm(n)*4*kt*50)+2*Bu(n)*w*Cbe_bc(n)*R(n)/(gm(n)*4*kt*50); 45 NF(n) = 10*log10(F(n)); C(n) = 1 + (w * Cbe_bc(n) * rb(n)).^2; Iout_icib(n) = twoq * Ic(n) + (twoq * Ib(n) + twoq * Ic(n) * (w * T(n)).^2) * gm(n).^2 * rb(n).^2/C(n) + 2 * Bu(n) * gm(n) * w * Cbe_bc(n) * rb(n).^2/C(n); Iout_rb(n) = 4*kt * rb(n) *gm(n).^2/C(n); Iout(n) = Iout_icib(n) + Iout_rb(n); Y21_inv(n) = (1 + j * w * Cbe_bc(n) * rb(n))/gm(n); Sva(n) = Iout(n)*(abs(Y21_inv(n))).^2; Rn(n) = Sva(n)/(4*kt); %%-----------------------------spice---------------------- Sib_spice(n) = twoq * Ib(n); Sic_spice(n) = twoq * Ic(n); Bu_spice(n) = 0; AA_spice(n) = Sic_spice(n) + Sib_spice(n) * wt_w(n).^2 + 2 * Bu_spice(n) * wt_w(n); Xopt_spice(n) = wCbe_inv(n) * (Sic_spice(n) + Bu_spice(n) * wt_w(n))/AA_spice(n);% - wCbe_inv(n); % "- wCbe_inv(n)" for LNA Ropt2_spice(n) = 4*kt * rb(n) * wt_w(n).^2/AA_spice(n) + rb(n).^2 + wCbe_inv(n).^2 * Sic_spice(n)/AA_spice(n) - wCbe_inv(n).^2 * (Sic_spice(n) + Bu_spice(n) * wt_w(n)).^2/(AA_spice(n).^2); Ropt_spice(n) = sqrt(Ropt2_spice(n)); Fmin_spice(n) = 1 + rb(n)/Ropt_spice(n) + 1/(4 * kt * Ropt_spice(n)) * (1/wt_w(n)).^2 * (Ropt_spice(n) + rb(n)).^2 * AA_spice(n) + 1/(4 * kt * Ropt_spice(n)) * wCbe_inv(n).^2 * (Sic_spice(n) * Sib_spice(n) - Bu_spice(n).^2) / AA_spice(n) ; NFmin_spice(n) = 10*log10(Fmin_spice(n)); R_spice(n) = (50+rb(n)).^2+0.^2; 46 F_spice(n) = 1+rb(n)/50+Sic_spice(n)*(1+(w*Cbe_bc(n)).^2*R_spice(n)- 2*w*Cbe_bc(n)*0)/(gm(n).^2*4*kt*50)+Sib_spice(n)*R_spice(n)/(4*kt*50)- 2*Bu_spice(n)*0/(gm(n)*4*kt*50)+2*Bu_spice(n)*w*Cbe_bc(n)*R_spice(n)/(gm(n)*4 *kt*50); NF_spice(n) = 10*log10(F_spice(n)); C(n) = 1 + (w * Cbe_bc(n) * rb(n)).^2; Iout_icib_spice(n) = twoq * Ic(n) + (twoq * Ib(n)) * gm(n).^2 * rb(n).^2/C(n) + 2 * Bu_spice(n) * gm(n) * w * Cbe_bc(n) * rb(n).^2/C(n); Iout_rb(n) = 4*kt * rb(n) *gm(n).^2/C(n); Iout_spice(n) = Iout_icib_spice(n) + Iout_rb(n); Y21_inv(n) = (1 + j * w * Cbe_bc(n) * rb(n))/gm(n); Sva_spice(n) = Iout_spice(n)*(abs(Y21_inv(n))).^2; Rn_spice(n) = Sva_spice(n)/(4*kt); end figure(1); subplot(4,1,1); hold on; plot(Jc*1e3,NFmin,'r-','LineWidth',2); plot(Jc*1e3,NFmin_spice,'r--','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('NF^{Device}_{min} (dB)'); subplot(4,1,2); hold on; plot(Jc*1e3,Ropt,'r-','LineWidth',2); plot(Jc*1e3,Ropt_spice,'r--','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('R^{Device}_{opt} (ohm)'); subplot(4,1,3); hold on; plot(Jc*1e3,Xopt,'r-','LineWidth',2); plot(Jc*1e3,Xopt_spice,'r--','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('X^{Device}_{opt} (ohm)'); subplot(4,1,4); hold on; 47 plot(Jc*1e3,Rn./50,'r-','LineWidth',2); plot(Jc*1e3,Rn_spice./50,'r--','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('R^{Device}_{n}/50'); 48 Appendix B Matlab Code for Noise Parameters of Matched LNA close all; clear all; format long; datapath = 'C:\Users\Xiaojia\Documents\Iccap\Noise\Noise5PAE_Hicum\7WLNoise_wrk\export_d ata\'; OPinfo = sprintf('%s7WL_p24_OPinfo.csv',datapath); OPinfo_wt = sprintf('%s7WL_p24_OPinfo_wt.csv',datapath); [OP] = textread(OPinfo,'','delimiter',',','headerlines',46); %'Vbe','0Hz','1','OP_betadc','OP_ib','OP_ic','OP_rb','OP_Cbe','OP_taun' [OP_wt] = textread(OPinfo_wt,'','delimiter',',','headerlines',36); %'','','','wt_gm','','','wt_fT' [Ic] = OP(:,6)'; [rb] = OP(:,7)'; [Cbe] = OP(:,8)'; [beta] = OP(:,4)'; [taun] = OP(:,9)'; [wt] = OP_wt(:,7)'; freq = 10e9; Length_e0 = 1 * 20; Width_e = 0.24; Ae = Width_e * Length_e0; q = 1.602189e-19; twoq = 2 * q; w = 2 * pi * freq; T = 290; k = 1.380662e-23; 49 kt = k * T; rb_per_um = rb * Length_e0; Ic_per_um = Ic/Length_e0; Cbe_per_um = Cbe/Length_e0; m = size(Ic,2); for n = 1:m Jc(n) = Ic(n)/Ae; Ropt(n) = 100; Length_e(n) = 1; while (Ropt(n) >= 50) Length_e(n) = Length_e(n) + 0.01; rb_2(n) = rb_per_um(n) / Length_e(n); Ic_2(n) = Ic_per_um(n) * Length_e(n); Cbe_2(n) = Cbe_per_um(n) * Length_e(n); Ae_2(n) = Length_e(n) * Width_e; Ib(n) = Ic_2(n)/beta(n); gm(n) = wt(n)*Cbe_2(n); wCbe_inv(n) = 1/(w * Cbe_2(n)); wt_w(n) = wt(n)/w; fg1 = 0.757; T(n) = taun(n)*fg1; Sib(n) = twoq * Ib(n) + twoq * Ic_2(n) * (w * T(n)).^2; Sic(n) = twoq * Ic_2(n); Sicib(n) = -j * twoq * Ic_2(n) * w * T(n); Bu(n) = -twoq * Ic_2(n) * w * T(n); 50 AA(n) = Sic(n) + Sib(n) * wt_w(n).^2 + 2 * Bu(n) * wt_w(n); Ropt2(n) = 4*kt * rb_2(n) * wt_w(n).^2/AA(n) + rb_2(n).^2 + wCbe_inv(n).^2 * Sic(n)/AA(n) - wCbe_inv(n).^2 * (Sic(n) + Bu(n) * wt_w(n)).^2/(AA(n).^2); Ropt(n) = sqrt(Ropt2(n)); end Xopt(n) = wCbe_inv(n) * (Sic(n) + Bu(n) * wt_w(n))/AA(n) - wCbe_inv(n); Fmin(n) = 1 + rb_2(n)/Ropt(n) + 1/(4 * kt * Ropt(n)) * (1/wt_w(n)).^2 * (Ropt(n) + rb_2(n)).^2 * AA(n) + 1/(4 * kt * Ropt(n)) * wCbe_inv(n).^2 * (Sic(n) * Sib(n) - Bu(n).^2) / AA(n) ; NFmin(n) = 10*log10(Fmin(n)); R(n) = (50+rb_2(n)).^2+0.^2; F(n) = 1+rb_2(n)/50+Sic(n)*(1+(w*Cbe_2(n)).^2*R(n)- 2*w*Cbe_2(n)*0)/(gm(n).^2*4*kt*50)+Sib(n)*R(n)/(4*kt*50)- 2*Bu(n)*0/(gm(n)*4*kt*50)+2*Bu(n)*w*Cbe_2(n)*R(n)/(gm(n)*4*kt*50); NF(n) = 10*log10(F(n)); C(n) = 1 + (w * Cbe_2(n) * rb_2(n)).^2; Iout_icib(n) = twoq * Ic_2(n) + (twoq * Ib(n) + twoq * Ic(n) * (w * T(n)).^2) * gm(n).^2 * rb_2(n).^2/C(n) + 2 * Bu(n) * gm(n) * w * Cbe_2(n) * rb_2(n).^2/C(n); Iout_rb(n) = 4*kt * rb_2(n) *gm(n).^2/C(n); Iout(n) = Iout_icib(n) + Iout_rb(n); Y21_inv(n) = (1 + j * w * Cbe_2(n) * rb_2(n))/gm(n); Sva(n) = Iout(n)*(abs(Y21_inv(n))).^2; Rn(n) = Sva(n)/(4*kt); end 51 figure(1); subplot(2,2,1); hold on; plot(Jc*1e3,NFmin,'r-','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('NFmin(dB)'); subplot(2,2,2); hold on; plot(Jc*1e3,NF,'r-','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('NF(dB)'); subplot(2,2,3); hold on; plot(Jc*1e3,Ropt,'r-','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('Ropt(ohm)'); subplot(2,2,4); hold on; plot(Jc*1e3,Xopt,'r-','LineWidth',2); xlabel('Jc(mA/um^2)');ylabel('Xopt(ohm)');