Microscopic theory of quadrupolar ordering in TmTe

We have calculated the crystal electric field of TmTe (T>T_Q) and have obtained that the ground state of a Tm 4f hole is the $\Gamma_7$ doublet in agreement with Mossbauer experiments. We study the quadrupole interactions arising from quantum transitions of 4f holes of Tm. An effective attraction is found at the L point of the Brillouin zone, $\vec{q}_L$. Assuming that the quadrupolar condensation involves a single arm of $\vec{q}_L$ we show that there are two variants for quadrupole ordering which are described by the space groups C2/c and C2/m. The Landau free energy is derived in mean-field theory. The phase transition is of second order. The corresponding quadrupole order parameters are combinations of $T_{2g}$ and $E_g$ components. The obtained domain structure is in agreement with observations from neutron diffraction studies for TmTe. Calculated lattice distortions are found to be different for the two variants of quadrupole ordering. We suggest to measure lattice displacements in order to discriminate between those two structures.