  The Nd permanent magnet (Nd$_{2}$Fe$_{14}$B) is an indispensable material
used in modern energy conversion devices. The realization of high coercivity at
finite temperatures is a burning issue. One of the important ingredients for
controlling the coercive force is the surface property of magnetic grains. It
has been reported by first-principles studies that the Nd atoms in the first
(001) surface layer facing the vacuum have in-plane anisotropy perpendicular to
the $c$ axis, which may decrease the coercivity. Focusing on the surface
anisotropy effect on the coercivity, we examine the coercivity at zero and
finite temperatures by using an atomistic model reflecting the lattice
structure of the Nd magnet with a stochastic Landau-Lifshitz-Gilbert equation
method. We study general three cases, in which the Nd atoms in surface layers
have (1) no anisotropy, (2) in-plane anisotropy, and (3) reinforced anisotropy
for two types of surfaces, (001) and (100) surfaces. We find that in contrast
to the zero-temperature case, due to the thermal fluctuation effect, the
modification of only the first surface layer has little effect on the
coercivity at finite temperatures. However, the modification of a few layers
results in significant effects. We discuss the details of the dependence of the
coercivity on temperature, type of surface, and modified layer depth, and also
the features of domain growth in magnetization reversal.
