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A bandgap voltage reference is a temperature independent voltage reference circuit widely used in integrated circuits. It produces a fixed (constant) voltage irrespective of power supply variations, temperature changes and the loading on the device. Usually it has an output voltage around 1.25 V, close to the theoretical 1.22 eV bandgap of silicon at 0 K. This circuit concept was first published by David Hilbiber in 1964. Bob Widlar, Paul Brokaw and others followed up with other commercially successful versions.
The voltage difference between two p-n junctions (e.g. diodes), operated at different current densities, is used to generate a proportional to absolute temperature (PTAT) current in a first resistor. This current is used to generate a voltage in a second resistor. This voltage in turn is added to the voltage of one of the junctions (or a third one, in some implementations). The voltage across a diode operated at constant current, or here with a PTAT current, is complementary to absolute temperature (CTAT—reduces with increasing temperature), with approx. −2 mV/K. If the ratio between the first and second resistor is chosen properly, the first order effects of the temperature dependency of the diode and the PTAT current will cancel out. The resulting voltage is about 1.2–1.3 V, depending on the particular technology and circuit design, and is close to the theoretical 1.22 eV bandgap of silicon at 0 K. The remaining voltage change over the operating temperature of typical integrated circuits is on the order of a few millivolts. This temperature dependency has a typical parabolic residual behavior since the linear (first order) effects are chosen to cancel.
Because the output voltage is by definition fixed around 1.25 V for typical bandgap reference circuits, the minimum operating voltage is about 1.4 V, as in a CMOS circuit at least one drain-source voltage of a FET (field effect transistor) has to be added. Therefore, recent work concentrates on finding alternative solutions, in which for example currents are summed instead of voltages, resulting in a lower theoretical limit for the operating voltage (Banba, 1999).
Note that sometimes confusion arises when using the abbreviation CTAT, where the "C" is incorrectly taken to mean "constant" rather than "complementary". To avoid this confusion, although not in widespread use, the term constant with temperature (CWT) is sometimes used.
When summing a PTAT (Proportional to Absolute Temperature) and a CTAT (Complementary to Absolute Temperature) current, only the linear terms of current are compensated, while the higher-order terms are limiting the TD (Temperature Drift) of the BGR at around 20ppm/oC, over a temperature range of 100 oC. For this reason, in 2001, Malcovati  designed a circuit topology that can compensate high-order non-linearities, thus achieving an improved TD. This design used an improved version of Banba  topology and an analysis of base-emitter temperature effects that was performed by Tsividis in 1980. In 2012, Andreou   has further improved the high-order non-linear compensation by using a second opamp along with an additional resistor leg at the point where the two currents are summed up. This method enhanced further the curvature correction and achieved superior TD performance over a wider temperature range. In addition it achieved improved Line Regulation and lower Noise.
The other critical issue in design of bandgap references is power efficiency and size of circuit. As a bandage reference is generally based on BJT devices and resistors, the total size of circuit could be very large which makes it very expensive for IC design. Moreover, this type of circuits might consume a lot of power to reach to the desired noise and precision spec. Therefore power efficiency and optimal design of such circuits is an interesting research topic.