We study the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) quadratic and biquadratic spin-spin interactions of two magnetic impurities in twisted bilayer graphene away from the magic angle. We apply the Bistritzer-MacDonald model of two graphene layers twisted with respect to each other by a small angle. By reducing the model to the Dirac-type one with modified Fermi velocity, we derive expressions for the RKKY quadratic and biquadratic spin interactions using perturbation theory for the free energy. The biquadratic interaction is suppressed by a larger power of the interaction constant and decreases faster with the distance between impurities compared to the quadratic one. Nevertheless, due to the different period of the oscillations with the impurity separation distance, chemical potential, twist angle, and temperature, it is possible to fine-tune the system to the regime of dominating biquadratic interaction. The existence of such a fine-tuned regime might provide a promising opportunity to observe nonconventional spin ordering.

In gapped bilayer graphene, similarly to conventional semiconductors, Coulomb impurities (such as nitrogen donors) may determine the activation energy of its conductivity and provide low-temperature hopping conductivity. However, in spite of the importance of Coulomb impurities, nothing is known about their electron binding energy Eb in the presence of gates. To close this gap, we study numerically the electron binding energy Eb of a singly charged donor in BN-enveloped bilayer graphene with the top and bottom gates at distance d and gate-tunable gap 2Δ. We show that for 10<d<200nm and 1<Δ<70meV the ratio Eb/Δ changes from 0.4 to 1.4. The ratio Eb/Δ stays so close to unity because of the dominating role of the bilayer polarization screening which reduces the Coulomb potential well depth to values ∼Δ. Still, the ratio Eb/Δ somewhat decreases with growing Δ, faster at small Δ and slower at large Δ. On the other hand, Eb/Δ weakly grows with d, again faster at small Δ and slower at large Δ. We also studied the effect of trigonal warping and found only a small reduction of Eb/Δ.