Electric-field-independent spin-orbit-coupling gap in <i>h</i>-BN-encapsulated bilayer graphene

The weak spin-orbit coupling (SOC) in bilayer graphene (BLG) is essential for encoding spin qubits while bringing technical challenges for extracting the opened small SOC gap ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$ in experiments. Moreover, in addition to the intrinsic Kane-Mele term, extrinsic mechanisms also contribute to SOC in BLG, especially under experimental conditions including encapsulation of BLG with hexagonal boron nitride (h-BN) and applying an external out-of-plane electric displacement field D. Although measurements of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$ in h-BN-encapsulated BLG have been reported, the relatively large experimental variations and existing experimental controversy make it difficult to fully understand the physical origin of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$. Here, we report a combined experimental and theoretical study on ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$ in h-BN-encapsulated BLG. We use an averaging method to extract ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$ in gate-defined single quantum dot devices. Under D fields as large as 0.57--0.90 V/nm, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$ \ensuremath{\approx} 53.4--61.8 $\text{\ensuremath{\mu}}$eV is obtained from two devices. Benchmarked with values reported at lower D field regime, our results support a D-field-independent ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$. This behavior is confirmed by our first-principles calculations, based on which ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SO}}$ is found to be independent of D field, regardless of different h-BN/BLG/h-BN stacking configurations. Our calculations also suggest a weak proximity effect from h-BN, indicating that SOC in h-BN-encapsulated BLG is dominated by the intrinsic Kane-Mele mechanism. Our results offer insightful understanding of SOC in BLG, which benefits SOC engineering and spin manipulations in BLG.