Anisotropic Heisenberg ferromagnets in 2D and electrically tunable ordering

Despite colossal efforts, room-temperature ferromagnetism in two-dimension remains elusive due to the fundamental spin fluctuation in reduced dimension. Here, we report a dramatic electrical manipulation of magnetic ordering up to room temperature in the monolayers of CrI₃ and CrBr₃ within the first-principles Heisenberg XXZ model. We develop an anisotropic Heisenberg model with relativistic exchange interactions that are obtained from the first-principles calculations. We demonstrate the crucial importance of magnetic interactions beyond the first-neighbour. The exchange and anisotropic magnetic interactions are externally modulated by a gate-induced charge carrier doping that triggers a non-trivial phase diagram. High-temperature ferromagnetism is associated with a substantial increase in effective ferromagnetic exchange and overall magnetic anisotropy under experimentally attainable hole doping. These results present new possibilities in electrically controlled spintronic and magnetoelectric devices based on atomically thin crystals.