  Dzyaloshinskii-Moriya interaction (DMI) is investigated in a 2D ferromagnet
(FM) with spin-orbit interaction of Rashba type at finite temperatures. The FM
is described in the continuum limit by an effective $s$-$d$ model with
arbitrary dependence of spin-orbit coupling (SOC) and kinetic energy of
itinerant electrons on the absolute value of momentum. In the limit of weak
SOC, we derive a general expression for the DMI constant $D$ from a microscopic
analysis of the electronic grand potential. We compare $D$ with the exchange
stiffness $A$ and show that, to the leading order in small SOC strength
$\alpha_{\text{R}}$, the conventional relation $D=(4
m\alpha_{\text{R}}/\hbar)A$, in general, does not hold beyond the
Bychkov-Rashba model. Moreover, in this model, both $A$ and $D$ vanish at zero
temperature in the metal regime (i.e., when two spin sub-bands are partly
occupied). For nonparabolic bands or nonlinear Rashba coupling, these
coefficients are finite and acquire a nontrivial dependence on the chemical
potential that demonstrates the possibility to control the size and chirality
of magnetic textures by adjusting a gate voltage.
