  We consider a hole-doped semiconductor with a sharp boundary and study the
boundary spin accumulation in response to a charge current. First, we solve
exactly a single-particle quantum mechanics problem described by the isotropic
Luttinger model in half-space and construct an orthonormal basis for the
corresponding Hamiltonian. It is shown that the complete basis includes two
types of eigenstates. The first class of states contains conventional incident
and reflected waves, while the other class includes localized surface states.
Second, we consider a many-body system in the presence of a charge current
flowing parallel to the boundary. It is shown that the localized states
contribute to spin accumulation near the surface. We also show that the spin
density exhibits current-induced Friedel oscillations with three different
periods determined by the Fermi momenta of the light and heavy holes. We find
an exact asymptotic expression for the Friedel oscillations far from the
boundary. We also calculate numerically the spin density profile and compute
the total spin accumulation, which is defined as the integral of the spin
density in the direction perpendicular to the boundary. The total spin
accumulation is shown to fit very well the simple formula S ~(1 - m_L/m_H)^2,
where m_L and m_H are the light- and heavy-hole masses. The effects of disorder
are discussed. We estimate the spin relaxation time in the Luttinger model and
argue that spin physics cannot be described within the diffusion approximation.
