  The Berry-phase mediated valley-selected skew scattering in alpha-T3 lattices
is demonstrated. The interplay of Lorentz and Berry forces in position and
momentum spaces is revealed and analyzed. Many-body screening of the
electron-impurity interaction is taken into account to avoid overestimation of
back- and skew-scattering of electrons in the system. Triplet peak from skew
interactions at two valleys is found in near-vertical and near-horizontal
forward- and backward-scattering directions for small Berry phases and low
magnetic fields. Magnetic-field dependence in both non-equilibrium and
thermal-equilibrium currents is also presented for valley-dependent
longitudinal and transverse transports mediated by a Berry phase.
Mathematically, two Boltzmann moment equations are employed for computing
scattering-angle distributions of non-equilibrium skew currents by using
microscopic inverse energy- and momentum-relaxation times. Meanwhile, a
valley-dependent unbalanced thermal-equilibrium anomalous Hall current induced
by the Berry force in momentum space, due to different mobilities for two
valleys, is also computed for comparisons.
