As may be clear from the previous sections, the design of bandgap references al- ready concerns a lot of parameters when only ideal physical models are used for the base-emitter voltages. For practical bandgap references, the models describ- ing the behavior of the transistor introduce even more parameters. Therefore, it is good to know which parameters of the practical model dominate the behavior of the transistor in the case of bandgap reference design.
The Gummel and Poon model [32] as used in SPICE [33] is a well known model and often used for circuit design. Therefore this model is used here as the basis for the design of bandgap references. A minimum set of key parameters will be derived that describes the relation between the base-emitter voltage and the collector bias current.
The bulk resistors are not taken into account because it is possible to make their influence negligibly small, especially in the case of low current design. The Gummel and Poon model is reduced further to the effects that are relevant for the forward-biased transistor. The leakage currents are ignored too, because in modern IC processes these leakage currents are negligibly small [34]. When these leakage currents cannot be ignored, due to the process characteristics or due to the relatively high temperature at which the bandgap reference has to operate, these leakage currents can be taken into account by using the descriptions as given in [27].
Further, it is assumed that the transistor is biased far from high-level injec- tion (if not, see again [27]). The relevant part of the Gummel and Poon model that remains is given by the following equation [35]:
The parameters that are used and their meanings are listed in table 6.1. A further reduction is obtained when the transistor is biased such that
In that case the forward Early effect, modeled by can be ignored. In contrast to which is on the order of several volts, which cannot be ignored. For a given temperature, is known and is the temperature at which the parameters are extracted from measurements. Finally, equals 1. Thus for an accurate design of bandgap references four parameters need to be known accurately, describing the relation between the base-emitter voltage and the collector current (see table 6.2). These parameters are the key parameters for
6.3. THE BASIC FUNCTION 175
bandgap reference design. When other models are used, instead of the Gummel and Poon model, the corresponding parameters are found.
6.3.4.1 The bandgap energy
The output voltage of the bandgap reference is directly related to the bandgap energy. An error in the description of the bandgap energy compared to the actual bandgap energy is directly seen in the reference voltage. The method described uses a Taylor expansion of the function describing the bandgap en- ergy. Thus it is largely independent of which function is used for the bandgap energy. In this book the approximation as described by Varshni [30] is used as it is the most commonly used one. It should be noted that in SPICE [33] a first-order approximation of this model is used. However, as can be seen from equation (6.13), both the In-function and the bandgap energy are responsible for the second and higher-order temperature behavior of the base-emitter voltage; their contributions appear to be on the same order of magnitude. Therefore, with SPICE-based simulators, second and higher-order compensated bandgap references cannot be simulated accurately.
6.3.4.2 The saturation current and its temperature behavior
The saturation current is one of the parameters determining the base-emitter voltage of a transistor. Via this parameter, spread is introduced in the base- emitter voltage due to emitter-area variations and variations in the doping level or doping profile. In the Gummel and Poon model, models the order of the temperature behavior of the saturation current.
6.3.4.3 The reverse Early voltage
This parameter is the only one modeling a non-ideality with respect to the ideal physical model of the transistor. The reverse Early voltage can be a serious problem. It models the base-width modulation at the base-emitter junction. As the doping level of the base is much lower than the doping level of the emitter, the variation of the depletion layer is predominantly found in the base region, in contrast to the base-width modulation at the base-collector junction where the variation is mainly found in the collector region. The reverse Early voltage can easily be on the order of just a few volts, e.g. 4V, which gives a reduction of the collector current of approximately (see equation (6.43)):
For accurate circuit design, therefore, the reverse Early effect also has to be taken into account. The error due to the reverse Early voltage in the output
176 CHAPTER 6. BANDGAP REFERENCES
voltage of the reference source is derived in [36]. It is given by:
This error is comparable to errors introduced by the spread in base-emitter voltages due to processing, et cetera [36]. Clearly, in contrast to what the name
reverse Early voltage suggests, this parameter is also important for the forward
mode of the transistor.