Where does a quantum dot engine run best?

A single-level dot between a hot lead and a cold lead turns heat into electrical work. Maximize its power and the efficiency usually lands at half of Carnot. Hold the bias fixed and tune only the gate, and it climbs to almost all of Carnot. Move the sliders to see why.

Efficiency at maximum power

Gate and bias, Eq. (18) Bias only, Eq. (22) Gate only, Eq. (40) ηC ηC/2 Curzon–Ahlborn
Near equilibrium the violet and teal curves leave the origin with slope ½, the textbook result for tightly coupled engines. The orange curve leaves with slope 1: with the bias pinned, the best gate setting sits almost at the reversible point.

Power landscape

Power over dot level EQD (across) and bias Δμ (up). The engine only runs below the diagonal Δμ = ηCEQD, where it becomes reversible. The line shows the slice you are allowed to move along.

Power along each slice

Power against η/ηC, each curve scaled to its own peak. Tuning the bias gives a near-parabola peaking at ½. Tuning the gate gives a curve that stays flat near zero efficiency and peaks close to 1.

Why the half-Carnot rule breaks

Van den Broeck's argument assumes the work-producing flux grows linearly with its force as you leave the stall point. Power is flux times force, so it is a parabola between zero force and the reversible point, and the peak sits halfway, at ηC/2.

With the bias fixed, the natural force is 1/EQD. Its zero, an infinitely high dot level, is not equilibrium: the current vanishes because the dot is simply out of reach, while the temperature difference is still there. Near that point the current dies like e−EQD/T2, an essential singularity rather than a straight line. If the flux instead rises like Xn+1, the peak moves to

ηop = ηC (n + 1)/(n + 2),

Peak at η/ηC = .

An essential singularity is the limit n → ∞, which is why gate-only tuning approaches Carnot efficiency at maximum power. The price is output: at ηC ≈ 0.3 and Δμ = T2 it delivers about 70% of the globally optimal power while running about 30% more efficiently, close to what Josefsson et al. measured.

Based on Sang Hoon Lee, Jaegon Um and Hyunggyu Park, “Nonuniversality of heat-engine efficiency at maximum power,” Phys. Rev. E 98, 052137 (2018). Units: kB = 1, T2 = 1, T1 = T2/(1 − ηC), tunneling rates normalized to 1. All curves are computed numerically in your browser.

Interactive demo created by Claude Opus 5.5 (Anthropic).