{c}

For $H_R=\lambda(k_x\sigma_y-k_y\sigma_x)$ with $\lambda>0$, the lower- and upper-band <spinors> of unit <norm> can be chosen as
$$
u_-\!=\frac1{\sqrt2}\begin{pmatrix}(k_y+ik_x)/k\\1\end{pmatrix},\qquad u_+\!=\frac1{\sqrt2}\begin{pmatrix}-(k_y+ik_x)/k\\1\end{pmatrix},\qquad k>0.
$$
Their <spin expectation values> are $\langle\mathbf S\rangle_\pm=\pm(\hbar/2)(-k_y,k_x,0)/k$. They are tangential to <circles> of constant <momentum> magnitude, opposite between branches, and reversed at opposite momenta as required by <time-reversal symmetry>. When $E_F>0$, the outer contour belongs to the lower branch and the inner one to the upper branch. For $E_F<0$ the two <Fermi contours> both belong to the lower branch and have the same tangential <spin> orientation at a given angle in <polar coordinates>. At $k=0$ the <Hamiltonian> has <quantum degeneracy> in <spin>, and no unique <spin texture> direction is selected.


 Rashba spin texture (source code)