  The intrinsic spin-Hall effect in hexagon-shaped samples is investigated. To
take into account the spin-orbit couplings and to fit the hexagon edges, we
derive the triangular version of the tight-binding model for the linear Rashba
[Sov. Phys. Solid State 2, 1109 (1960)] and Dresselhaus [Phys. Rev. 100, 580
(1955)] [001] Hamiltonians, which allow direct application of the
Landauer-Keldysh non-equilibrium Green function formalism to calculating the
local spin density within the hexagonal sample. Focusing on the out-of-plane
component of spin, we obtain the geometry-dependent spin-Hall accumulation
patterns, which are sensitive to not only the sample size, the spin-orbit
coupling strength, the bias strength, but also the lead configurations.
Contrary to the rectangular samples, the accumulation pattern can be very
different in our hexagonal samples. Our present work provides a fundamental
description of the geometry effect on the intrinsic spin-Hall effect, taking
the hexagon as the specific case. Moreover, broken spin-Hall symmetry due to
the coexistence of the Rashba and Dresselhaus couplings is also discussed. Upon
exchanging the two coupling strengths, the accumulation pattern is reversed,
confirming the earlier predicted sign change in spin-Hall conductivity.
