Quantum Dot Lab

A single-level quantum dot between two leads: the smallest system in which electron–electron interactions shape both charge and heat transport. Lower the dot level with the gate to add electrons one at a time, change the Coulomb interaction U or the temperature T and watch Coulomb blockade and the thermopower respond, or heat one lead to drive a thermoelectric current.

Energy diagram · hot source (red) and cold drain (blue) filled up to their chemical potentials μL = +eV/2 and μR = −eV/2, each smeared by its own temperature · dot levels εd and εd+U shaded by their spectral weight · arrow: electron flow
Occupation
Conductance (V → 0)
Thermopower (V → 0)
ZTel (V → 0)
Current I
Power P = −IV · efficiency
Transmission 𝒯(ω) at the chosen εd · shaded: transport window (−∂f/∂ω without bias or ΔT, otherwise |fL − fR|)
Conductance G (2e²/h) and occupation n (dashed, right axis) vs dot level εd · click to move εd
Thermopower S (kB/e) vs dot level εd · click to move εd
Current I (eΓ/h) vs bias at the chosen εd and ΔT · red: power delivered · click to move the bias
Heat engine: power P = −IV (Γ²/h) from V = 0 to the thermovoltage, where the current stops · click to move the bias

What you are seeing

  • Transmission. Interactions split the level into two peaks, at εd and εd+U. Their weights depend on the occupation: an electron of one spin either blocks the other spin or lets it in at the cost of U. Transport only happens inside the Fermi window.
  • Conductance. One peak for every electron added. Between the peaks the dot is Coulomb-blockaded, and in the middle it holds exactly one electron (n = 1). Raising T smears the peaks out.
  • Thermopower. It changes sign each time a transmission peak crosses the Fermi level: electron-like (S < 0) when transport happens above it, hole-like (S > 0) below. At the particle–hole symmetric point, εd = −U/2, it vanishes.
  • Thermal gradient. With the source hotter than the drain, a current flows even without any voltage. Above μ the hot lead has more electrons, so they flow from hot to cold; below μ it has more empty states, so electrons flow from cold to hot (holes go hot → cold). The direction flips exactly where the thermopower changes sign.
  • Bias and heat engine. A bias opens a window between μL and μR; the current jumps each time εd or εd+U enters it (Coulomb steps). With a hot source, electrons keep flowing even against a small opposing bias: the dot turns heat into electrical power, P = −IV > 0 (red region), with an efficiency η bounded by Carnot, ηC = 1 − TR/TL. The bias at which the current stops is the thermovoltage.

Model: spin-degenerate Anderson impurity in the Coulomb-blockade regime, with the equation-of-motion (Hubbard-I) spectral function solved self-consistently for the occupation, and symmetric coupling ΓL = ΓR = Γ/2. G, S and ZT are linear-response quantities at zero bias and the mean temperature; the current, power and efficiency are computed at finite bias and ΔT, beyond linear response, with the occupation evaluated out of equilibrium using the Fermi functions of both leads. Kondo correlations (T < TK) are not included. Energies in units of Γ. For an analytic equation-of-motion treatment of the double-dot case, see Sobrino, Jacob & Kurth, Phys. Rev. B 110, 115121 (2024) and Phys. Rev. B 111, 115108 (2025).