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 f_T and f_max.

Fig. 1 shows measured I_B-V_BE characteristics at several temperatures, together with simulation using the latest version of Mextram, 505.3. V_CB=0V. At moderate V_BE, the I_B-V_BE slope starts to decrease, similar to high injection effect in diodes, when we know high injection has not occurred as I­_C-V_BE 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 I_B-V_BE behavior [5]. In this work we focus on the low V_BE regions. At V_BE<0.2V, I_B first increases, then decreases, exhibiting a negative differential resistance, which is due to BTBT [3]. This BTBT region shows very weak temperature dependence. I_B 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 I_B2 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: I_B,BTBT = k_BTBTV_BE,btbt(V_BE,btbt – V_BTBT)², K_BTBT and V_BTBT are model parameters that can be extracted from the I_B peak position and slope of I_B-V_BE followed by optimization. V_BTBT has a physical meaning of V_BTBT =[(E_fn-E_c)|_n-side+(E_v-E_fp)|_p-side]/q, with all symbols having their usual meanings in PN junctions. V_BE,btbt is a smoothly limited version of V_BE for modeling BTBT. When V_BE<0, V_BE is smoothly limited to 0, i.e. V_BE,btbt->0, so that it does not interfere with the existing reverse bias BTBT model. When 0<V_BE< V_BTBT, V_BE,btbt is essentially equal to V_BE by design except at the boundaries. When V_BE>V_BTBT, V_BE is smoothly limited to V_BTBT, 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 I_TAT = I_STAT(exp(V_BE,tat/V_TUN)-1), with V_TUN and I_STAT as model parameters [4]. V_BE,tat is approximately V_BE when V_BE>0, and smoothly becomes zero when V_BE<0. I_STAT scales with temperature T according to I_STAT,T=I_STAT(T/T_ref)^1/2exp(k_TAT(T-T_ref)), where T_ref is the reference temperature, and k_TAT is a temperature scaling model parameter. V_TUN is temperature independent.

Fig. 2 shows the 25°C I_B-V_BE modeling result at low V_BE using a linear current scale, to better illustrate the behavior of all I_B components at low V_BE. 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 I_B-V_BE 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.