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We have formulated a theory for investigating the conditions which arerequired to achieve entangled states of electrons on graphene andthree-dimensional (3D) topological insulators (TIs). We consider the quantumentanglement of spins by calculating the exchange energy. A gap is opened up atthe Fermi level between the valence and conduction bands in the absence ofdoping when graphene as well as 3D TIs are irradiated with circularly-polarizedlight. This energy band gap is dependent on the intensity and frequency of theapplied electromagnetic field. The electron-photon coupling also gives rise toa unique energy dispersion of the dressed states which is different from eithergraphene or the conventional two-dimensional electron gas (2DEG). In ourcalculations, we obtained the dynamical polarization function for imaginaryfrequencies which is then employed to determine the exchange energy. Thepolarization function is obtained with the use of both the energy eigenstatesand the overlap of pseudo-spin wave functions. We have concluded that whiledoping has a significant influence on the exchange energy and consequently onthe entanglement, the gap of the energy dispersions affects the exchangeslightly, which could be used as a good technique to tune and controlentanglement for quantum information purposes.

Theory for entanglement of electrons dressed with circularly polarized light in Graphene and three-dimensional topological insulators