X-Parameters Based Characterization and Compact Modeling of SiGe HBT Linearity
Authors: Niu Guofu , Zhang Anni , Li Yiao , Zhang Huaiyuan , Scholten Andries , Willemsen Marnix , Pijper Ralf , Tiemeijer L.F. 
Published: December 2020

Linearity is an important SiGe HBT RF characteristics that has received a great deal of research for decades. For relatively small signal RF circuits such as LNAs, linearity is often characterized by measuring 3rd order intercept points, IP3, using a spectrum analyzer, or better, a Nonlinear Vector Network Analyzer (NVNA). For highly linear common-base SiGe HBTs, however, we find that the distortions generated within the RF source, which itself is also a transistor circuit, can be amplified by the HBT under test, which then interfere with the HBT’s distortions, making experimental validation of distortion in compact modeling difficult, or even impossible, for biases of high IP3. The devices used are the high voltage SiGe HBTs from NXP’s QUBIC4Xi technology. We overcome this problem with X-parameter (an invention of Keysight) measurements that are uniquely capable of removing the unwanted RF source harmonics as well as correcting for source mismatch [1], leading to more accurate transistor IP3 measurement. Fig. 1 shows the fundamental and 3rd order Pout versus Pin measured on an on-wafer through structure, with and without using X-parameters. Using X-parameters, the 3rd order harmonic output power is significantly reduced, increasing measurement system’s OIP3 to about 50 dBm.
X-parameters approximate the generally nonlinear relationship between reflected waves and incident waves at the fundamental and harmonic frequencies, by a linear expansion around the system response excited by a relatively large fundamental stimulus at port 1, a situation of practical interest for amplifiers and mixers, as shown in Fig. 2. Nonlinear effects are relatively weak so that harmonic components can be superposed. The reflected wave at port p and harmonic index k, Bpk
, relates to the large incident wave A11
, and all the small signal incident waves aql
(port k, harmonic index l) by Eqs. (1) and (2) embedded in Fig. 2. The term P is the phase of the large signal fundamental incident wave. The superscript “*” means taking complex conjugate. N is number of ports, and K is number of harmonics. In our measurements, N=2, K=3.
Fig. 3 shows output IP3 (OIP3) versus VCB for IE=1, 10, 20 and 30 mA, calculated from two FB type X-parameters, XFB23 and XFB21. Fundamental frequency is 2GHz. The Mextram 505.1 model does an excellent job in modeling IP3. While IP3 is an important linearity figures-of-merit, just like fT
and fmax
are for current and power gains, IP3 is not a complete representation of transistor linearity, just like fT
and fmax
represent only current and power gain for specific circuit terminations. At a given bias and fundamental frequency, there are for instance 78 X-parameters, for up to order 3 harmonics, including FB, S and T types, while only two FB type X-parameters are used for IP3. A logic question is if our compact model, in this case, Mextram, is capable of accurately modeling the complete X-parameter set, which represent distortion behavior much more completely. A systematic comparison of measured and modeled X-parameters is thus called for. Unfortunately, X-parameters are not supported yet by ICCAP, an industry standard device modeling tool also from Keysight, due to the complexity of X-parameter simulation. Out of necessity, here we use ADS to simulate X-parameters and then manually overlay simulation results with measurement by post-processing, a tedious process, particularly when model parameters are tuned. To a large extent, Mextram 505.1 [2], the latest version, is found to accurately model nearly all the X-parameters for a quite large range of emitter current and collector voltage biasing conditions.
As an example, we show in Fig. 4 the FB and S type X-parameters as a function of VCB, for an IE of 30mA. A11 corresponds to an input power of -5dBm. The T type X-parameters for this current are small and below noise floor of measurement, and thus not shown. In the full paper we will also discuss the importance of modeling avalanche, particularly its biasing current dependence as it relates to Kirk’s effect, in modeling X-parameters, as well as how to use behavior of specific X-parameters to identify the dominant distortion mechanism, e.g. CB capacitance nonlinearity, avalanche nonlinearity, or transport current nonlinearity.
[1] D. Root, J. Verspecht, J. Horn, and M. Marcu, X-Parameters, Characterization, Modeling and Design of Nonlinear RF and Microwave Components, Cambridge Press, 2013.
[2] G. Niu et al., The Mextram Bipolar Transistor Model, version 505.1.0, Auburn University, 2019.

Figure 1