Microwave spectroscopy of few-carrier states in bilayer graphene quantum dots

TL;DR

This paper demonstrates high-resolution microwave spectroscopy of few-carrier states in bilayer graphene quantum dots using a high-impedance cQED resonator. By detecting dispersive shifts of interdot transitions, the authors resolve zero-field Kane-Mele spin-orbit gaps and observe Pauli spin and valley blockades for two- and three-carrier configurations, achieving energy resolutions far surpassing traditional transport. The results elucidate the detailed spin-valley structure of BLG DQDs, reveal state-selective coupling via dipole transitions, and show potential for fast, dispersive qubit readout of Kramers singlet-triplet qubits in BLG. This technique provides a powerful platform for probing spin-valley physics and informs the design of BLG-based qubits with enhanced control and scalability.

Abstract

Bilayer graphene is a maturing material platform for gate-defined quantum dots that hosts long-lived spin and valley states. Implementing solid-state qubits in bilayer graphene requires a fundamental understanding of such confined electronic systems. In particular, states of two and three carriers, for which the exchange interaction between particles plays a crucial role, are a cornerstone for qubit readout and manipulation. Here we report on the spectroscopy of few-carrier states in bilayer graphene quantum dots, using circuit quantum electrodynamics (cQED) techniques that offer substantially improved energy resolution compared to standard transport techniques. Measurements using a superconducting high-impedance resonator capacitively coupled to the double quantum dot reveal dispersive features of two and three electron states, enabling the detection of Pauli spin and valley blockade and the characterization of the spin-orbit gap at zero magnetic field. The results deepen our understanding of few-carrier spin and valley states in bilayer graphene quantum dots and demonstrate that cQED techniques are a powerful state-selective probe for semiconductor nanostructures.

Figures (9)

• Figure 1: (a) Single-carrier states in BLG are labeled by a valley and spin. Spin-orbit coupling $\Delta_\mathrm{SO}$ separates the four possible states into two degenerate Kramers pairs. States with parallel alignment of valley and spin magnetic moments (symbolized by arrows) are lowered in energy, while anti-parallel alignment raises their energy. (b) Two-carrier DQD states at the $(1,1) \leftrightarrow (0,2)$ interdot transition. The valley-polarized $(1,1)$ ground state is incompatible with the valley-singlet $(0,2)$ ground state, leading to Pauli blockade. (c) Device schematic of the hybrid device. A high-impedance resonator is probed through a feedline in a notch-type geometry. Its center conductor is connected to the left plunger gate. The DQD device is fabricated from a van der Waals hetero-structure with BLG encapsulated in hBN. (d) Energy diagram of the two-carrier DQD states as a function of detuning $\delta$ with finite tunneling coupling $t$ that hybridizes the electronic states. Their transition dipole moment interacts dispersively with the microwave resonator of energy $h f_\mathrm{r}<2t$.
• Figure 2: Pauli blockade at the $(1,1) \leftrightarrow (0,2)$ charge transition measured in direct transport and in the microwave response of the resonator. (a) In the forward direction with $V_\mathrm{SD} > 0$, current is large within the finite bias triangles. (b) In the blocked direction with $V_\mathrm{SD} < 0$, current is suppressed by more than two orders of magnitude due to selection rules of the involved ground states. (c) The microwave response shows three distinct features at the base of the triangles. A high-resolution measurement of the outlined region is presented in Figure \ref{['fig2']} (a). (d) The blockade is also detected in the microwave response as the signal from the interdot transition is absent.
• Figure 3: Finite-bias resonator response at the $(1,1) \leftrightarrow (0,2)$ [panels (a) - (d)] and $(1,2) \leftrightarrow (0,3)$ [panels (e) - (h)] transitions. The microwave transmission $S_{21}$ at $V_\mathrm{SD} = +0.5mV$ shows distinct features at finite DQD detunings. (a) The $n=2$ resonator signal is explained qualitatively by two low-lying excited states of the $(1,1)$ charge configuration, labeled as $L^\prime$ and $L^{\prime \prime}$. (ii) Detuning dependence of the two-carrier DQD states with finite tunnel coupling. Panel (iii) shows a line cut along the detuning axis marked in (i) (b) Finite source-drain bias creates a non-equilibrium population of excited states if the hybridized charge states (dashed lines) fall within the bias window. Visibility of the excited state transitions is therefore enhanced inside the lower bias triangle. (c) Transitions at finite detuning energies indicated by teal, purple and yellow correspond to the arrows of same color in (a). (d) Detuning energies of excited state transitions extracted from fits to the signal peaks. (e) - (h) The $n=3$ transition similarly shows three distinct features in $S_{21}$. However, now the central feature (teal arrow) corresponds to the $L \leftrightarrow R$ ground-state transition, as evident from its high visibility in the region between lower and upper triangle. The observed signal is explained by the presence of one low-lying excited state in either charge configuration, i.e. $L^\prime$ and $R^\prime$.
• Figure 4: Schematic of interdot transitions between $(1,n-1)$ and $(0,n)$ charge configurations for $n=2,3$. (a) Two-carrier states in a DQD. With negligible exchange interaction for the $(1,1)$ charge configuration, the 16 possible spin-valley states split into three bundles, separated in energy by $\Delta_\mathrm{SO}$. Their coupling to the $(0,2)$ ground state, a valley-singlet spin-triplet, gives rise to the dispersive signature presented in Figure \ref{['fig3']} (a). (b) Three-carrier interdot transition. Both the $(1,2)$ and $(0,3)$ charge configuration feature one excited state, separated from the ground-state by $\Delta_\mathrm{SO}$. In contrast to $n=1$, the corresponding excited-state transitions (purple arrows) are not blocked and lead to dispersive resonator interaction at finite positive and negative detuning values [see Figure \ref{['fig3']} (e)].
• Figure 5: State-selective dipole coupling of a Kramer's singlet-triplet qubit. Energy dispersion of the $(1,1) \leftrightarrow (0,2)$ interdot transition as a function of detuning energy $\delta$. A small perpendicular magnetic field splits spin- and valley polarized states off the unpolarized state (teal and green lines). The valley-singlet spin-triplet $(0,2)$ ground state $\ket{S_\mathrm{v} T^0_\mathrm{s}}_{(0,2)}$ couples only to the anti-symmetric Kramers-singlet $\ket{A}_{(1,1)}$, resulting in a finite transition dipole sensed by the on-chip microwave resonator. The symmetric Kramers-triplet $\ket{B_0}_{(1,1)}$ on the other hand does not couple.