Physics Quantum Dot Charging

Quantum dot charging

A quantum dot is a nanoscale conductive island, often a few tens of nanometers across, connected to leads by thin tunnel barriers. Because its capacitance C is extremely small, adding a single electron of charge e changes its electrostatic energy by E_C = e²/(2C), a quantity called the charging energy. When this energy exceeds the thermal energy k_B T, electrons cannot pass through the dot one at a time without paying that price; the current is suppressed at low bias voltages. This effect, the Coulomb blockade, is the working principle behind single-electron transistors and a standard tool for measuring the charge of a single electron.

The blockade condition is usually written k_B T << E_C, meaning the thermal energy must be far below the charging energy. In 1998, the framework's cost function J(x) = (x + 1/x)/2 - 1, which measures the forced price of any ratio x, enters this condition through a specific threshold. The framework models the blockade as observable when E_C > J(φ) k_B T, where φ is the golden ratio. Since J(φ) = φ - 3/2 ≈ 0.118, this gives the explicit regime k_B T < E_C / 0.118 ≈ 8.47 E_C. That is a concrete numerical bound: the thermal energy must be less than about 8.5 times the charging energy, a condition consistent with the experimental rule of thumb that k_B T must be much smaller than E_C.

In the framework's machine-checked library of formal theorems, the module for this topic proves three general facts about the cost function, not about quantum dots specifically. It proves that the cost vanishes when the two quantities in the ratio are equal, that the cost is never negative for positive inputs, and that the threshold φ - 3/2 is positive. These are universal properties of J, stated once in a shared template and instantiated here with the ratio m/e, where m and e are placeholders for any two positive real numbers. The module does not define what m and e mean for a real quantum dot; that step, connecting the abstract ratio to a physical charge and energy, remains open.

The practical consequence is a clean, checkable bridge between the framework's cost function and a real experimental phenomenon. The Coulomb blockade is not derived from first principles here; rather, the framework supplies a candidate threshold that matches the known experimental condition. The three proved facts are the load-bearing pieces: they guarantee the cost is a sensible, nonnegative measure that is zero when the two quantities balance, and they fix the golden-ratio threshold as a positive number. What the module does not do is identify m and e with specific physical quantities, so the numerical match to the 8.47 E_C regime is a model choice, not a theorem.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove the Coulomb blockade condition from physical first principles. The numerical match to 8.47 E_C is a model choice, not a derived theorem. No specific physical meaning for m and e is established in this module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/QuantumDot_Charging.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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