A standard bandgap reference sums a CTAT voltage (V_BE, roughly -2 mV/°C) with a PTAT voltage (ΔV_BE scaled up, roughly +2 mV/°C) to produce ~1.25 V that's flat with temperature. But "flat" is a lie — the CTAT term isn't purely linear. V_BE(T) has a logarithmic term that leaves a parabolic residual of about 2–3 mV curvature across -40 to +125°C. That's why a plain Brokaw or Kuijk bandgap tops out around 25–50 ppm/°C. To reach the 1–5 ppm/°C you see in an LTC6655 or ADR445, you need curvature correction.

The V_BE equation reveals the villain:

V_BE(T) = V_G0 − (V_G0 − V_BE0)(T/T₀) − (η − x)(kT/q)ln(T₀/T)

The first two terms cancel against a PTAT voltage. That trailing T·ln(T) term is what a first-order bandgap can't kill. It's why the reference output curves downward at both temperature extremes and peaks in the middle.

Three practical correction techniques:

• PTAT² correction: Add a current proportional to T² (generated by squaring a PTAT current in a translinear loop) into the output summing node. Cancels the parabolic residual directly. Used in the classic LT1019.
• Piecewise-linear correction: Turn on an extra PTAT current only above a threshold temperature using a comparator. Cheap in silicon area, gets you to ~10 ppm/°C. Common in mid-tier references.
• V_BE-loop correction (Brokaw's trick): Run two transistors at different current densities and different temperature coefficients of collector current. The V_BE difference contains a term that cancels the log-nonlinearity. Elegant, but hard to trim.

Real-world example: The ADR441 uses PTAT² curvature correction plus laser-trimmed thin-film resistors to hit 3 ppm/°C over -40 to +85°C. Compare a hobbyist-grade LM4040 at 100 ppm/°C — that's a 33× improvement, and it's why a 16-bit ADC front end costs $8 instead of $0.50.

Rule of thumb: A first-order bandgap's uncorrected curvature is roughly ΔV ≈ 3 mV peak-to-peak across the full industrial range. On a 2.5 V reference, that's 3 mV / 2.5 V / 165°C ≈ 7 ppm/°C average, but the peak slope near the endpoints is 3–5× worse. If your system trims at 25°C and operates at 85°C, budget at least 20 ppm/°C of drift error from an uncorrected reference — usually the dominant error in a 14-bit-plus signal chain.

Don't forget: curvature correction only fixes the deterministic parabola. Long-term drift (10–50 ppm/√kh from package stress relaxation) and 1/f noise are separate battles.