  Two-dimensional (2D) intrinsic ferromagnetic (FM) semiconductors are crucial
to develop low-dimensional spintronic devices. Using density functional theory,
we show that single-layer chromium trihalides (SLCTs) (CrX$_3$,X=F, Cl, Br and
I) constitute a series of stable 2D intrinsic FM semiconductors. A
free-standing SLCT can be easily exfoliated from the bulk crystal, due to a low
cleavage energy and a high in-plane stiffness. Electronic structure
calculations using the HSE06 functional indicate that both bulk and
single-layer CrX$_3$ are half semiconductors with indirect gaps and their
valence bands and conduction bands are fully spin-polarized in the same spin
direction. The energy gaps and absorption edges of CrBr$_3$ and CrI$_3$ are
found to be in the visible frequency range, which implies possible
opt-electronic applications. Furthermore, SLCTs are found to possess a large
magnetic moment of 3$\mu_B$ per formula unit and a sizable magnetic anisotropy
energy. The magnetic exchange constants of SLCTs are then extracted using the
Heisenberg spin Hamiltonian and the microscopic origins of the various exchange
interactions are analyzed. A competition between a near 90$^\circ$ FM
superexchange and a direct antiferromagnetic (AFM) exchange results in a FM
nearest-neighbour exchange interaction. The next and third nearest-neighbour
exchange interactions are found to be FM and AFM respectively and this can be
understood by the angle-dependent extended Cr-X-X-Cr superexchange interaction.
Moreover, the Curie temperatures of SLCTs are also predicted using Monte Carlo
simulations and the values can further increase by applying a biaxial tensile
strain. The unique combination of robust intrinsic ferromagnetism, half
semiconductivity and large magnetic anisotropy energies renders the SLCTs as
promising candidates for next-generation semiconductor spintronic applications.
