Elementary excitations, exchange interaction and spin-Peierls transition in CuGeO$_3$

The microscopic description of the spin-Peierls transition in pure and doped CuGeO_3 is developed taking into account realistic details of crystal structure. It it shown that the presence of side-groups (here Ge) strongly influences superexchange along Cu-O-Cu path, making it antiferromagnetic. Nearest-neighbour and next-nearest neighbour exchange constants $J_{nn}$ and $J_{nnn}$ are calculated. Si doping effectively segments the CuO_2-chains leading to $J_{nn}(Si)\simeq0$ or even slightly ferromagnetic. Strong sensitivity of the exchange constants to Cu-O-Cu and (Cu-O-Cu)-Ge angles may be responsible for the spin-Peierls transition itself (``bond-bending mechanism'' of the transition). The nature of excitations in the isolated and coupled spin-Peierls chains is studied and it is shown that topological excitations (solitons) play crucial role. Such solitons appear in particular in doped systems (Cu_{1-x}Zn_xGeO_3, CuGe_{1-x}Si_xO_3) which can explain the $T_{SP}(x)$ phase diagram.