Extended Data Fig. 5: Comparison between the ν = ±3 phase transitions in the control device C1 and the asymmetric device A1. | Nature

Extended Data Fig. 5: Comparison between the ν = ±3 phase transitions in the control device C1 and the asymmetric device A1.

From: Spin–orbit-driven band inversion in bilayer graphene by the van der Waals proximity effect

Extended Data Fig. 5

a, Energy level diagram of the zero-energy LL in the absence of SOC. The ν = ±3 transitions are occurring between ground states with identical spin polarization. Note that offsets from u* = 0 are possible due to differing on-site energies within the BLG unit cell, which can arise from coupling to the hBN substrate, but that these offsets do not influence the spin degree of freedom. The solid and dashed lines differentiate spin orientation. b, Measured \({D}_{\nu =\pm 3}^{\ast }\) as a function of the total magnetic field (BT) for fixed B = 4 T in control device C1. No Zeeman dependence is observed, consistent with expectations from an SOC-free model. The red dashed line is the average value of \({D}_{\nu =\pm 3}^{\ast }\). c, Energy level diagram of the zero-energy LL with a layer-selective Ising SOC of λI = 5 meV, with sign chosen so that the effect of the SOC opposes the external field (reproduced from Fig. 2f of the main text). Note that the ν = ±3 transitions now occur between ground states with opposite spin polarization. d, Measured \({D}_{\nu =\pm 3}^{\ast }\) as a function of BT for fixed B = 4 T in device A1, reproduced from the main text. The red dashed line is a two-parameter fit with λI = 1.7 meV and ϵBLG = 2.8, with the latter needed for the conversion between experimentally measured D and theoretically calculated u. e, Schematic of the effect of BT in an asymmetric device. The red curve plots the dot product of the spin orientation on the top layer and the magnetic field, and the blue curve plots the product of the spin orientation on the bottom layer and the magnetic field. Whereas the LL in the unaffected layer always aligns its spin polarization with the external magnetic field (see the red arrows in the dashed boxes for total external magnetic fields of 5, 10 and 20 T, respectively), the spin polarization of LLs in the SOC-proximitized bottom layer result from a competition between SOC-induced Zeeman field (out of plane) and the changing direction of the physical Zeeman field (see the blue arrows in the dashed boxes). The affected spin cants only slightly for \({E}_{{\rm{Z}}}\ll {\lambda }_{{\rm{I}}}\), but eventually the Zeeman energy overwhelms the SOC and the two spins align as EZ/λI → ∞.

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