Spin-valley coupling in single-electron bilayer graphene quantum dots

ARTICLE https://doi.org/10.1038/s41467-021-25498-3 OPEN Spin-valley coupling in single-electron bilayer graphene quantum dots 1234567890():,; L. Banszerus T. Taniguchi 1,2 ✉, S. Möller 5, C. Volk 1,2, C. Steiner1,2, E. Icking1,2, S. Trellenkamp3, F. Lentz3, K. Watanabe 1,2 & C. Stampfer 4, 1,2 Understanding how the electron spin is coupled to orbital degrees of freedom, such as a valley degree of freedom in solid-state systems, is central to applications in spin-based electronics and quantum computation. Recent developments in the preparation of electrostatically-confined quantum dots in gapped bilayer graphene (BLG) enable to study the low-energy single-electron spectra in BLG quantum dots, which is crucial for potential spin and spin-valley qubit operations. Here, we present the observation of the spin-valley coupling in bilayer graphene quantum dots in the single-electron regime. By making use of highly-tunable double quantum dot devices we achieve an energy resolution allowing us to resolve the lifting of the fourfold spin and valley degeneracy by a Kane-Mele type spin-orbit coupling of ≈ 60 μeV. Furthermore, we find an upper limit of a potentially disorder-induced mixing of the K and K 0 states below 20 μeV. 1 JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, Aachen, Germany. 2 Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, Jülich, Germany. 3 Helmholtz Nano Facility, Forschungszentrum Jülich, Jülich, Germany. 4 Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan. 5 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. ✉email: NATURE COMMUNICATIONS | (2021)12:5250 | https://doi.org/10.1038/s41467-021-25498-3 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-25498-3 T he valley pseudospin is an inherent property of twodimensional honeycomb crystals and - together with the electron spin - makes graphene and bilayer graphene (BLG) interesting for applications in spin- and valley-based electronics and quantum computation1,2. This pseudospin arises from the orbital degree of freedom of the independent energy valleys located at the inequivalent vertices (K and K 0 ) of the hexagonal Brillouin zone3. In analogy to the real spin, the valley pseudospin exhibits also a valley Zeeman effect4–6, where the valley Zeeman splitting – varying linearly with (out-of-plane) magnetic field – is a result of the orbital magnetic moments originating from the nonvanishing Berry curvature, Ω, at the K-points of gapped BLG (see Fig. 1a). Since these magnetic moments, which have opposite signs for the two valleys, crucially depend on the wave function, the valley g-factor in BLG quantum dots can be tuned by electric fields7,8, offering promising and interesting possibilities for manipulation. However, to fully exploit the potential to manipulate and control both the valley and spin degrees of freedom in BLG quantum dots (QDs), a detailed understanding of their a interaction is essential. This is as relevant for a better understanding of spin decoherence processes as it is for exploring ways to electrically manipulate the spin degree of freedom via spin-orbit interaction and implementing innovative spin-valley qubits2. Indeed, a detailed understanding of the low-energy spectrum of single particle states within the first electronic orbital (see Fig. 1b) is crucial for finding suitable working points and manipulation mechanisms for possible qubit operation. Although the single-particle spectrum in BLG QDs has been intensively studied in recent years9–11, the low-energy spin-valley coupling in BLG QDs has remained experimentally unexplored. This is certainly partly due to the high energy resolution required, as theoretical studies predict an intrinsic spin-orbit (SO) coupling in graphene and BLG of around ΔSO ≈ 24 μeV12–16 and only recently, experiments have – partly indirectly – reported values in the range between 40 and 80 μeV17,18. Moreover, our current knowledge with respect to a possible mixing of K and K 0 states is very limited. The latter is expressed by ΔKK0 and could allow to access helical states19. E E K´ K Ω K´ Ω K E b K K Δ KK´ Δ SO K´ K´ B B Fig. 1 Band structure and single particle spectrum of a BLG quantum dot. a Low energy band schematic of BLG at the K and K 0 points. BLG exhibits a nontrivial Berry curvature Ω that leads to an effective out-of-plane magnetic moment with opposite sign at K and K 0 . b Energy dispersion of single-particle states in BLG QDs as a function of in-plane (B∥, left) and out-of-plane (B⊥, right) applied magnetic fields with respect to the BLG plane. The SO gap, ΔSO, lifts the fourfold degeneracy and polarizes the spins out-of-plane for zero magnetic field and a potential K-K 0 state mixing (described by ΔKK0 ) leads to an anticrossing of the K # and K 0 # state. 2 NATURE COMMUNICATIONS | (2021)12:5250 | https://doi.org/10.1038/s41467-021-25498-3 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-25498-3 a d GL GC GR 4.1 GL GC GR SG (1,1) 4.0 BLG (0,1) (0,0) hBN 4.0 Drain Source QD L 140 VGR (V) 70nm b I (pA) 0 QD R c EF 4.1 VGL (V) GL p QD L GC 4.2 GR QD R p Fig. 2 Device tuning. a False-color scanning electron micrograph of the metallic gates. A pair of split gates defines the conducting channel, which can be modulated by voltages applied to the finger gates. The gates used in the following are color coded. b Schematic cross-section of the device. The upper part shows the metallic gates on top of the hBN/BLG/hBN van-der-Waals heterostructure. The lower part color codes the charge carrier density within the channel and the two quantum dots (red: holes and blue: electrons). c Schematics of the band edge profile along the narrow channel, highlighting how the finger gates are used to form a DQD consisting of QDL and QDR connected to the p-type conducting channel. d Charge stability diagram showing the current through a double quantum dot as function of the potential applied to the two gate fingers, VGL and VGR. A constant bias voltage of Vb = 1 mV is applied and the central finger gate voltage is kept at VGC = −4 V. In this letter, we report on measurements of the excited state spectrum of single-electron double quantum dots (DQDs) in BLG providing information on ΔSO as well as on ΔKK0 . By tuning a DQD to a regime of low interdot tunnel coupling, we are able to resolve the interdot transitions with remarkably high energy resolution allowing to reconstruct the underlying single particle spectrum of both quantum dots. We find that the spin and valley degeneracy of the single particle spectrum is lifted by a Kane–Mele type SO gap13 of ΔSO ≈ 60 μeV, which separates the two Kramer’s pairs – ðK 0 "; K #Þ and ðK 0 #; K "Þ – similar (but smaller in magnitude) to what (...truncated)