Entropy spectroscopy of a bilayer graphene quantum dot

TL;DR

The paper demonstrates Maxwell relation–based entropy spectroscopy as a direct probe of ground-state degeneracy in a gate-defined bilayer graphene quantum dot. By thermally modulating the reservoir and monitoring the second-harmonic charge detector response, the authors extract the entropy changes during single- and double-electron transitions, revealing a two-fold degeneracy for the one-carrier ground state and a nondegenerate two-carrier ground state at zero field. The nondegenerate two-carrier state is attributed to Kane–Mele type spin–orbit coupling mixing valley and spin degrees of freedom, with finite-bias spectroscopy and extracted g-factors supporting this picture. This entropy-based method provides a powerful, complementary tool to transport spectroscopy for characterizing ground-state degeneracies in graphene and other van der Waals heterostructures, with potential applications to exotic quantum states and qubit technologies.

Abstract

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.

Figures (17)

• Figure 1: (a) The system consists of a quantum dot (QD) thermally coupled to a reservoir. The reservoir temperature $T(t)$ is changed by driving a current $I_\mathrm{heater}$ through an ohmic region in the BLG. This region is induced by the heater structure, consisting split gates (SG) and finger gates (FG). The charge detector (CD), capacitively coupled to the QD, carries a current $I_\mathrm{det}$ which changes as the number of charge carriers changes on the QD. For the $0\to 1$- and $1\to 2$-transition, respectively: The DC component of the detector current (b) and (e) with the extracted temperature modulation (continuous traces are fits to eq. \ref{['eq:detDC']}, the electronic background temperatures are $\overline{T}_{0\to 1} = 60mK$ and $\overline{T}_{1\to 2} = 95mK$), second harmonic current normalized to $I_0$ (c) and (f) with extracted entropy change (continuous traces are fits to eq. \ref{['eq:dethamtwo']}), and the entropy obtained by integration (d) and (g) (continuous traces are plots of eq. \ref{['eq:S01']} in (d) and eq. \ref{['eq:S12']}supplemental2025 in (g)). The gray bars $\pm 0.1 \ln(2)$ as a guide to the eye.
• Figure 2: Evolution of the measured entropy change in out-of-plane magnetic field together with the calculated entropy change (solid line) for the $0\to 1$ (a) and $1\to 2$ transition (b). for the $0\to 1$ and $1\to 2$ transitions. Finite bias spectroscopy measurements in out-of-plane magnetic field performed for the $0\to 1$ (c) and $1\to 2$ transition (d). The states are identified by their behavior in out-of-plane magnetic field.
• Figure 3: Excited state spectra for the one-carrier and two-carrier charge state in an out-of-plane magnetic field including the effect of Kane-Mele type spin--orbit interaction for the $N=2$ charge state. The ground state $\ket{S_v T_s^0} \oplus \ket{T_v^0 S_s}$ is separated from the spin-triplet states $\ket{S_v T_s^+}$, $\ket{S_v T_s^-}$ by an energy gap $\Delta_\mathrm{SO}'$. The ground state crossing is due to the valley-triplet state $\ket{T_v^- S_s}$ and appears at an out-of-plane magnetic field $B^\times$.
• Figure S1: (a) Optical microscope image of the used sample. The BLG flake is outlined in white. The applied back gate voltage is negative, rendering the BLG p-type/hole conducting. The split gate pairs SGM, SGB and SGM, SGT are used to electrostatically define a conducting channel. (b) Schematic of the VdW stack structure in the region where the QD is defined: BLG is encapsulated by hexagonal boron nitride (hBN) surrounded by a global graphite back gate ($\mathrm{BG}$) and with two layers of gold top gates: split gates SGM, SGB to form a one-dimensional channel, and finger gates to form the QD (plunger gate PG and tunneling barrier gates LB, RB). The top gate layers are separated by aluminum oxide ($\mathrm{AlOx}$) as a dielectric. The displacement field is roughly $-0.46V\per nm$ resulting in a band gap of approximately $50meV$ickingTransportSpectroscopyUltraclean2022. The device is operated in the hole conduction regime. The barriers are tuned such that there is no coupling to the left lead. AFM images of the (c) QD (black circles) formed with finger gates LB, PG, RB and the charge detector tuned with finger gate T, and the (d) (e) heater gate structures to form the ohmic constrictions, which are formed with split gates SGM, SGC1, SGC2 and finger gates FGC1 and FGC2.
• Figure S2: (a) Charge detector current as a function of left and right barrier voltage. The plunger gate voltage is fixed to $3.25V$. Three types of resonances, corresponding to three types of QDs are observed: 1. pn-junction defined electron QD under LB corresponding to the horizontal transconductance peaks. 2. barrier defined hole QD between LB and RB corresponding to the bent transconductance peaks. This is the QD used for the entropy measurements. 3. pn-junction defined electron QD under RB corresponding to the vertical transconductance peaks. The number triplet labels the occupation number of the respective QDS (1., 2. 3.). (b) Diagrams showing the band bending induced by the finger gates LB, PG, RB in order to form the different types of QDs for a few exemplary cases. QDs form where the conduction band edge $E_c$ or valence band edge $E_v$ crosses the chemical potential $\mu$ in energy.