  We investigated the dependence of N$\acute{\textrm{e}}$el temperature as a
critical temperature on the electric field and hole-electron doping in the
antiferromagnetically ordered zigzag graphene nanoribbon. The temperature was
calculated by averaging the magnon energy in the Brillouin zone within the
mean-field approximation. We employed the generalized Bloch theorem instead of
the supercell approach to reduce the computational cost significantly to obtain
the magnon spectrum. We showed that the N$\acute{\textrm{e}}$el temperature
reduces when increasing both the electric field and the hole-electron doping,
thus these treatments will never enhance the N$\acute{\textrm{e}}$el
temperature.
