Entropy spectroscopy of a bilayer graphene quantum dot

We measure the entropy change of charge transitions in an electrostatically defined quantum dot in bilayer graphene. Entropy provides insights into the equilibrium thermodynamic properties of both ground and excited states beyond transport measurements. For the one-carrier regime, the obtained entropy shows that the ground state has a two-fold degeneracy lifted by an out-of-plane magnetic field. This observation is in agreement with previous direct transport measurements and confirms the applicability of this novel method. For the two-carrier regime, the extracted entropy indicates a non-degenerate ground state at zero magnetic field, contrary to previous studies suggesting a three-fold degeneracy. We attribute the degeneracy lifting to the effect of Kane-Mele type spin–orbit interaction on the two-carrier ground state, which has not been observed before. Our work demonstrates the validity and efficacy of entropy measurements as a unique, supplementary experimental tool to investigate the degeneracy of the ground state in quantum devices build in materials such as graphene. This technique, applied to exotic systems with fractional ground state entropies, will be a powerful tool in the study of quantum matter.

💡 Research Summary

In this work the authors present a thermodynamic study of an electrostatically defined quantum dot (QD) in bilayer graphene (BLG) by measuring the entropy change associated with single‑electron charge transitions. The experimental platform consists of a plunger gate that controls the addition energy µ_N, a pair of on‑chip heaters that generate a low‑frequency temperature modulation of the electron reservoir, and a nearby charge detector (CD) whose current responds to the QD occupation. By driving the heater with an AC current at frequency ω = 40 Hz, the reservoir temperature oscillates at 2ω, producing a second‑harmonic component I_2ω^det in the detector current. This component is directly proportional to the temperature derivative of the average occupation, (∂N/∂T)_µ,T, which, through the Maxwell relation S(µ,T) – S(µ_0,T) = ∫_µ0^µ (∂N/∂T)_µ,T dµ, yields the entropy change ΔS between adjacent charge states.

Two complementary analysis routes are employed. Method A assumes the thermally broadened regime (k_B T ≫ Γ) and fits the analytical expression for (∂N/∂T) derived from the Fermi function to the measured I_2ω^det, extracting ΔS as a fitting parameter without an independent temperature calibration. Method B first calibrates the temperature modulation ΔT via separate Joule‑heating measurements, then uses the proportionality I_2ω^det = I_0 (∂N/∂T) ΔT to obtain (∂N/∂T) directly and integrates it to obtain ΔS. Both methods give consistent results for the 0 → 1 and 1 → 2 electron transitions.

For the 0 → 1 transition the entropy rises from zero to a maximum of k_B ln 3 and then settles at k_B ln 2, indicating a two‑fold degenerate ground state (Kramers pair |K⁻↓⟩, |K⁺↑⟩) at zero magnetic field. Applying an out‑of‑plane magnetic field B⊥ lifts this degeneracy via combined spin and valley Zeeman effects. The measured evolution of ΔS with B⊥ matches a model that uses a valley g‑factor g_v ≈ 13.6, a spin g‑factor g_s ≈ 2.0, and a Kane‑Mele spin‑orbit gap Δ_SO ≈ 75 µeV, all in agreement with earlier transport spectroscopy on similar devices.

The 1 → 2 transition reveals a strikingly different behavior. The entropy change is –k_B ln 2, and the final entropy of the N = 2 charge state is essentially zero, demonstrating a non‑degenerate ground state at B = 0 T. This contradicts previous transport studies that inferred a three‑fold degenerate valley‑singlet spin‑triplet ground state. The magnetic‑field dependence shows that up to ~100 mT the entropy change follows the lifting of the one‑electron Kramers degeneracy, while at B× ≈ 220 mT a sharp increase of ΔS by k_B ln 2 occurs. This is interpreted as a ground‑state crossing: the N = 2 ground state, originally a superposition |S_v T_0 s⟩ ⊕ |T_0 v S_s⟩ (a mixture of valley‑singlet spin‑triplet and valley‑triplet spin‑singlet configurations), is overtaken by an excited valley‑triplet spin‑singlet state |T⁻_v S_s⟩ whose energy decreases with B⊥. The crossing produces the observed entropy peak.

To rationalize these observations the authors introduce a four‑electron Hamiltonian that incorporates the Kane‑Mele type intrinsic spin‑orbit interaction. This interaction couples the two N = 2 configurations, lifting the three‑fold degeneracy and splitting the levels into a non‑degenerate ground state and higher‑lying spin‑triplet states separated by an additional spin‑orbit gap Δ′_SO. The model reproduces the measured g‑factors (the second electron’s valley g‑factor g_v ≈ 15.5, larger than that of the first electron due to confinement‑induced renormalization) and the field‑dependent energy spectrum, confirming that the intrinsic spin‑orbit coupling is responsible for the observed degeneracy lifting.

Overall, the study demonstrates that entropy spectroscopy provides a direct, equilibrium‑based probe of quantum‑dot level degeneracies, complementary to conventional finite‑bias transport spectroscopy which infers degeneracies indirectly from excited‑state splittings. In BLG quantum dots, where spin, valley, and orbital degrees of freedom intertwine, entropy measurements can reveal subtle effects such as the Kane‑Mele spin‑orbit interaction that are otherwise difficult to resolve. The authors suggest that extending this technique to systems with fractional or non‑Abelian ground‑state entropies (e.g., ν = 5/2 fractional quantum Hall states, Majorana platforms, multi‑channel Kondo devices) could open a powerful new avenue for exploring exotic quantum matter.