Abstract
Particle–hole symmetry plays an important role in the characterization of topological phases in solid-state systems1. It is found, for example, in free-fermion systems at half filling and it is closely related to the notion of antiparticles in relativistic field theories2. In the low-energy limit, graphene is a prime example of a gapless particle–hole symmetric system described by an effective Dirac equation3,4 in which topological phases can be understood by studying ways to open a gap by preserving (or breaking) symmetries5,6. An important example is the intrinsic Kane–Mele spin-orbit gap of graphene, which leads to a lifting of the spin-valley degeneracy and renders graphene a topological insulator in a quantum spin Hall phase7 while preserving particle–hole symmetry. Here we show that bilayer graphene allows the realization of electron–hole double quantum dots that exhibit near-perfect particle–hole symmetry, in which transport occurs via the creation and annihilation of single electron–hole pairs with opposite quantum numbers. Moreover, we show that particle–hole symmetric spin and valley textures lead to a protected single-particle spin-valley blockade. The latter will allow robust spin-to-charge and valley-to-charge conversion, which are essential for the operation of spin and valley qubits.
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Data availability
The data supporting the findings are available in a Zenodo repository under accession code https://doi.org/10.5281/zenodo.7821944.
Code availability
The simulation code is available in a Zenodo repository under accession code https://doi.org/10.5281/zenodo.7821944
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Acknowledgements
We thank H. Bluhm, K. Flensberg, F. Haupt and L. Schreiber for helpful discussions, and F. Lentz, S. Trellenkamp and D. Neumaier for help with sample fabrication. This project received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 881603 (Graphene Flagship); from the European Research Council (grant agreement no. 820254) from Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 0 390534769, through DFG (STA 1146/11-1); and from the Helmholtz Nano Facility54. K.W. and T.T. acknowledge support from JSPS KAKENHI (grant nos. 19H05790, 20H00354 and 21H05233).
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L.B., C.V. and C.S. conceived the experiment. L.B., S.M., K.H. and E.I. fabricated the device. L.B., S.M. and C.V. performed measurements and analysed data. S.M. and F.H. performed simulations of the current. K.W. and T.T. synthesized hBN crystals. C.V. and C.S. supervised the project. L.B., S.M., C.V., F.H. and C.S. wrote the manuscript with contributions from all authors. L.B. and S.M. contributed equally to this work.
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Extended data figures and tables
Extended Data Fig. 1 Charge stability diagrams for opposite bias voltages in DQD #1.
a,b, Charge stability diagrams of DQD #1 (as in Fig. 1d) measured at a bias voltage of VSD = 1 mV (a) and VSD = −1 mV (b) (T = 10 mK). The dashed circles mark the formation of single electron – single hole DQDs using the hole QD and an electron QD to the left (DQD #3, red) or right (DQD #1, black) of the hole QD. c,d, Schematics of the valence and conduction band edge profiles along the p-type channel. An electron-hole double quantum dot is formed using the hole QD and the electron QD underneath the left (right) FG (see red (black) circles in Extended Data Figs. 1a,b).
Extended Data Fig. 2 Extracting ΔSO from measurements on a single-electron DQD in the same device.
a, Charge stability diagrams of the (1e, 0e) ↔ (0e, 1e) transition of an electron-electron DQD measured at VSD = 1 mV and B⊥ = 0 T (T = 10 mK). A ground state and an excited state transition are visible (see black arrows). b, Cut along the yellow dashed line in a. Two Lorentzian peaks (dashed lines) are fitted to the data. Inset, schematic energy diagrams of an electron-electron DQD in the finite bias regime for different interdot detuning energies ε, illustrating resonant transport from the left (L) to the right (R) QD through the ground state of each QD (transition (i)) and resonant transport at ε = ΔSO (transition (ii)). Data taken from ref. 10.
Extended Data Fig. 3 Additional data set for another electron-hole double quantum dot (DQD #2) in the same device.
a,b, Gate configurations used to form the DQDs #1, #3 and #2 in the device, respectively. c, Charge stability diagram of an e-h DQD formed with the second set of gate fingers (DQD #2, see panel b). The dashed circle marks the (0h, 0e) → (1h, 1e) transition. VSD = 1 mV (T = 10 mK). d,e, Close-ups of the (0h, 0e) → (1h, 1e) triple point measured at VSD = 0.5 mV and VSD = 1.5 mV, respectively. Transport only occurs via the α and β transition. f, Charge stability diagram as in panel e measured at B⊥ = 0.6 T. g, Charge stability diagram as in panel e at B∥ = 0.7 T. h,i, Charge stability diagrams as in panels d,e at VSD = − 0.5 mV and VSD = −1.5 mV. Transport is strongly suppressed; only co-tunnelling can be observed. j,k, Charge stability diagrams as in panels f,g measured at B⊥ = 0.6 T and B∥ = 0.7 T and VSD = −1.5 mV.
Extended Data Fig. 4 Probing the single-particle particle-hole symmetric spectrum.
a, Energy dispersion of single-particle states in the first orbital for electrons and holes as a function of in-plane (B∥, left) and out-of-plane (B⊥, right) magnetic fields. States and transitions are labelled as in Fig. 3a. b, Current through DQD #2 as a function of the detuning energy \(\widetilde{\varepsilon }\) (see yellow dashed line in Extended Data Fig. 3e) and B⊥ at VSD = 1.5 mV. The white dashed line marks the onset of the bias transport window. c, Current through the device as a function of \(\widetilde{\varepsilon }\) and B∥ at VSD = 1.5 mV. d,e, Data acquired in the blockade regime (VSD = −1.5 mV). The current has been measured as a function of B⊥ and B∥, respectively. Data have been symmetrized around B = 0.
Extended Data Fig. 5 Charge stability diagrams of the first triple point simulated by solving the rate equation.
a–c, The forward bias direction (VSD = 1 mV) for different magnetic fields, showing the same features as the experimental data presented in Fig. 2. d–f, The blocked bias direction (VSD = − 1 mV) for the same magnetic fields. For zero magnetic field, the blockade is lifted at the corners of the bias triangle, where back and forth tunneling to source (or drain) allows lifting the blockade. The effect is even larger at finite parallel magnetic fields, where the spins are tilted into the plane of the BLG.
Extended Data Fig. 6 Comparison of the single particle electron-hole blockade and singlet-triplet Pauli blockade.
a,b, Schematic of the chemical potentials in an electron-hole DQD for a ϵ = 0 and b ϵ = Δorb. For clearity the energy axis of the schematic is drawn to scale, such that Eg = 10Δorb = 100ΔSO. It can be seen that the band gap energy protects the blockade to be lifted by detuning. c,d, The same configurations in a unipolar DQD in (1,1) → (2,0) configuration. The singlet-triplet splitting ΔS-T is chosen to be equal to the orbital splitting Δorb in a,b. Is can be seen that the blockade is lifted as soon as ϵ = ΔS-T. Additionally, it can be lifted by relaxation of the triplet (1,1) state into the singlet (1,1) state.
Extended Data Fig. 7 Transport properties simulated for asymmetric valley g-factors in the electron and hole QDs.
a–h, Calculation of the current through the device as a function of the detuning energy \(\widetilde{\varepsilon }\) (see arrow in Fig. 2c) and perpendicular magnetic field at a finite bias of VSD = 2 mV (a–d) and VSD = −2 mV (e–h). In a, the valley g-factors of the two QDs are chosen asymmetrically (\({g}_{{\rm{v}}}^{{\rm{e}}}=15\) for the electron QD and \({g}_{{\rm{v}}}^{{\rm{h}}}=20\) for the hole QD), resulting in a splitting of both, the α and β transition, which scales with the difference in the valley g-factors. In b, the valley g-factors of the two QDs are chosen less asymmetrically (\({g}_{{\rm{v}}}^{{\rm{e}}}=15\) for the electron QD and \({g}_{{\rm{v}}}^{{\rm{h}}}=17\) for the hole QD), resulting in a smaller splitting of both, the α and β transition, which scales with the difference in the valley g-factors. In c the valley g-factors are chosen symmetrically (gv = 15), and no dependence on B⊥ is observed. In d, the experimentally observed g-factor difference of \({g}_{{\rm{v}}}^{{\rm{e}}}=15\) and \({g}_{{\rm{v}}}^{{\rm{h}}}=15.1\) is used for the simulation. e–h, For reverse bias, the single-particle blockade remains robust and the current is zero, independent of the chosen valley g-factor asymmetry, as the spin and valley texture, that is, the level ordering remains symmetrical.
Extended Data Fig. 8 Determining the g-factor asymmetry.
a, Exemplary line trace of the tunnelling current at B⊥ = 0 as a function of the detuning. The sum of two Gauss curves with width Γ is fitted to the data (see dashed line). b, Γ extracted from the line fits as shown in a, as a function of B⊥. Attributing the linear broadening of α and β to an asymmetry of valley g-factors between electron and hole QD yields Δg ≈ 0.11.
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Banszerus, L., Möller, S., Hecker, K. et al. Particle–hole symmetry protects spin-valley blockade in graphene quantum dots. Nature 618, 51–56 (2023). https://doi.org/10.1038/s41586-023-05953-5
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DOI: https://doi.org/10.1038/s41586-023-05953-5
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