Compact Modeling of Forward Operation Band-to-Band andTrap-Assisted Tunneling Currents in SiGe HBTs
Authors: Zhang Anni , Niu Guofu , Li Yiao , Scholten Andries 
Published: November 2022

Band-to-band tunneling (BTBT) [1] and trap-assisted tunneling (TAT) [2] have been observed in the forward operation base current of SiGe HBTs [3], due to increasing emitter-base junction doping and decreasing bandgap from high Ge mole fraction. At present, a compact model for the BTBT base current in forward operation does not exist. The reverse bias BTBT current is modeled in Mextram for both emitter-base and collector-base junctions. A compact model of the forward TAT base current was developed by our group for wide temperature range operation [4]. In this work, we present a new compact model of BTBT and further develop the TAT compact model of [4] for commercial temperature range application, and demonstrate their effectiveness on SiGe HBTs with around 250 GHz peak fT and fmax.
Fig. 1 shows measured IB-VBE characteristics at several temperatures, together with simulation using the latest version of Mextram, 505.3. VCB=0V. At moderate VBE, the IB-VBE slope starts to decrease, similar to high injection effect in diodes, when we know high injection has not occurred as IC-VBE slope remains its low injection value (not shown). Such behavior cannot be modeled with standard base current models. A newly developed neutral-base-recombination (NBR) model introduced in 505.3 is used to fit such high-injection like IB-VBE behavior [5]. In this work we focus on the low VBE regions. At VBE<0.2V, IB first increases, then decreases, exhibiting a negative differential resistance, which is due to BTBT [3]. This BTBT region shows very weak temperature dependence. IB then slowly increases, with a large non-ideality factor of 3.78, which we believe is due to TAT [2] [3] [4]. Note that this should not be confused with the depletion-region recombination current, called IB2 in Mextram [5], with an ideality factor close to 2. The corresponding saturation current in this “TAT” region shows a temperature dependence much weaker than what one expects from the depletion region recombination current as well.
Our BTBT model is based on the approach of Esaki [1] and Karlovsky [6], and has a simple functional form: IB,BTBT = kBTBTVBE,btbt(VBE,btbt – VBTBT)2, KBTBT and VBTBT are model parameters that can be extracted from the IB peak position and slope of IB-VBE followed by optimization. VBTBT has a physical meaning of VBTBT =[(Efn-Ec)|n-side+(Ev-Efp)|p-side]/q, with all symbols having their usual meanings in PN junctions. VBE,btbt is a smoothly limited version of VBE for modeling BTBT. When VBE<0, VBE is smoothly limited to 0, i.e. VBE,btbt->0, so that it does not interfere with the existing reverse bias BTBT model. When 0<VBE< VBTBT, VBE,btbt is essentially equal to VBE by design except at the boundaries. When VBE>VBTBT, VBE is smoothly limited to VBTBT, so that the BTBT current decays to zero, as there will be no aligned available states for BTBT. We observe that the BTBT current is insensitive to temperature, consistent with [3], and find that no temperature scaling is necessary for modeling the BTBT current.
The TAT current is modeled as ITAT = ISTAT(exp(VBE,tat/VTUN)-1), with VTUN and ISTAT as model parameters [4]. VBE,tat is approximately VBE when VBE>0, and smoothly becomes zero when VBE<0. ISTAT scales with temperature T according to ISTAT,T=ISTAT(T/Tref)1/2exp(kTAT(T-Tref)), where Tref is the reference temperature, and kTAT is a temperature scaling model parameter. VTUN is temperature independent.
Fig. 2 shows the 25°C IB-VBE modeling result at low VBE using a linear current scale, to better illustrate the behavior of all IB components at low VBE. Fig. 3 shows a full range view on a logarithmic scale. Fig. 4 shows the modeling results at all four temperatures, with the insert showing the low bias details.
Overall the proposed BTBT and TAT models do a good job fitting the forward operation IB-VBE characteristics, with an easy to use formulation, and a small number of model parameters, 2 for BTBT and 3 for TAT.

References

[1] L. Esaki, Phys. Rev., vol. 109, pp. 604–605, Jan. 1958.
[2] A. G. Chynoweth et al., Phys. Rev., vol. 121, pp.684-694, Feb. 1961.
[3] D. Lagarde et al., IEEE Electron Device Letters, vol. 27, no. 4, pp. 275-277, April 2006.
[4] Z. Xu et al., ECS Transactions, vol. 33, no. 6, pp. 301-310, 2010.
[5] G. Niu et al., The Mextram Bipolar Transistor Model Version 505.3.0, 2022.

[6] J. Karlovsky, Phys. Rev., vol. 127, pp. 419–419, July 1962.

Figure 1